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Slide 1 / 183
Geometry
Congruent Triangles
2015-10-23
www.njctl.org
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Table of Contents· Congruent Triangles
· Isosceles Triangle Theorem
· SSS Congruence· SAS Congruence· ASA Congruence · AAS Congruence· HL Congruence
· CPCTC· Triangle Congruence Proofs
click on the topic to go to that section
· Proving Congruence
· PARCC Sample Questions
Slide 3 / 183
Throughout this unit, the Standards for Mathematical Practice are used.
MP1: Making sense of problems & persevere in solving them.MP2: Reason abstractly & quantitatively.MP3: Construct viable arguments and critique the reasoning of others. MP4: Model with mathematics.MP5: Use appropriate tools strategically.MP6: Attend to precision.MP7: Look for & make use of structure.MP8: Look for & express regularity in repeated reasoning.
Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used.
If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab.
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Throughout this unit, the Standards for Mathematical Practice are used.
MP1: Making sense of problems & persevere in solving them.MP2: Reason abstractly & quantitatively.MP3: Construct viable arguments and critique the reasoning of others. MP4: Model with mathematics.MP5: Use appropriate tools strategically.MP6: Attend to precision.MP7: Look for & make use of structure.MP8: Look for & express regularity in repeated reasoning.
Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used.
If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab.
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Mat
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Congruent Triangles
Return to Tableof Contents
Slide 5 / 183
Similar Triangles
We learned in the Similar Triangles topic (Triangles unit) that if two triangles are similar:
· All their angles are congruent· All their corresponding sides are in proportion
We also learned how to identify the corresponding sides as being opposite to equal angles, or subtended by equal angles
And we learned that the constant of proportionality for the corresponding sides of one triangle to the other was called "k."
If needed, go back to review that topic before proceeding.
Slide 6 / 183
Congruent Triangles
Congruent triangles are a special case of similar triangles.
The constant of proportionality is one, so the corresponding sides are of equal measure.
For congruent triangles, all the angles are congruent AND all the corresponding sides are congruent.
Slide 7 / 183
Naming Congruent Triangles
Just as in the case of similar triangles, the naming of congruent triangles is important: order matters.
The statement:ΔABC is congruent to ΔDEF
indicates that these triangles are congruent.
AND that these angle measures are equal: m∠A = m∠D m∠B = m∠E m∠C = m∠F
AND these lengths are equal:
AB = DEBC = EFCA = FD
Slide 8 / 183
Naming Congruent Triangles
Just as in the case of similar triangles, the naming of congruent triangles is important: order matters.
The statement:ΔABC is congruent to ΔDEF
indicates that these triangles are congruent.
AND that these angle measures are equal: m∠A = m∠D m∠B = m∠E m∠C = m∠F
AND these lengths are equal:
AB = DEBC = EFCA = FD
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Mat
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Make sure that the students understand that the order in which you name the triangles matters. This slide explains why.
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Proving Triangles Congruent
We can prove triangles congruent by proving the measures of all three corresponding angles and the lengths of all three corresponding sides are equal.
Earlier we showed that we need to prove only two angles are congruent to show that triangles are similar, since the third angle must then be congruent.
There are similar shortcuts to proving triangles congruent.
Slide 9 / 183
Third Angle TheoremRecall the proof showing if we know that two pairs of corresponding angles are congruent, then the third pair of corresponding angles are congruent as well.
Statement Reason
1 ∠A ≅ ∠D and ∠B ≅ ∠E Given
2 m∠A = m∠D; m∠B = m∠E Definition of ≅ angles
3 m∠A+ m∠B + m∠C = 180ºm∠D+ m∠E + m∠F = 180º
Triangle Sum Theorem
4 m∠D+ m∠E + m∠C = 180ºm∠D+ m∠E + m∠F = 180º
Substitution Property of Equality
5 m∠D + m∠E + m∠C = m∠D + m∠E + m∠F
Substitution Property of Equality
6 m∠C = m∠F Subtraction Property of Equality
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Corresponding Parts
Let's review identifying the corresponding parts (angles and sides) of pairs of triangles.
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Given that ΔABC is congruent to ΔDEF, identify all the congruent corresponding parts
A
B
C
D
E
F
Corresponding Parts
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Given that ΔABC is congruent to ΔDEF, identify all the congruent corresponding parts
A
B
C
D
E
F
Corresponding Parts
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Ans
wer
∠A ≅ ∠D
∠B ≅ ∠E
∠C ≅ ∠F
AB ≅ DE
BC ≅ EF
CA ≅ FD
This example addresses MP6.
Slide 12 (Answer) / 183
A
B
C
D
E
F
Given that ΔABC is congruent to ΔDEF, the triangles are marked accordingly in this diagram.
Corresponding Parts
Slide 13 / 183
A
B
D
C
E
Part Corresponding PartSegment AB Segment ED
∠A ∠E
Segment AC Segment EC
∠B ∠D
Segment CB Segment CD
∠ACB ∠ECD
ΔABC ≅ ΔEDC
Slide 14 / 183
Example
Corresponding Sides Corresponding Angles
Given that ΔABC ≅ ΔLMN, identify all the corresponding angles and sides. (Draw a diagram)
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Example
Corresponding Sides Corresponding Angles
Given that ΔABC ≅ ΔLMN, identify all the corresponding angles and sides. (Draw a diagram)
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Teac
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otes
Corresponding SidesAB LMBC MNAC LN
=~=~=~
Corresponding Angles
Have students arrive at the answers as a class, or independently.
This example addresses MP5 & MP6
Additional Q's that address MP standards:How could you start this problem? (MP1)How could you use a drawing to assist with this problem? (MP5)How can you make sure that your answer is accurate? (MP6)
Slide 15 (Answer) / 183
1 What is the corresponding part to ∠ J ? A ∠R B ∠K C ∠Q D ∠P
J
K L R Q
P
ΔJKL ≅ ΔPQR
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1 What is the corresponding part to ∠ J ? A ∠R B ∠K C ∠Q D ∠P
J
K L R Q
P
ΔJKL ≅ ΔPQR
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Ans
wer
D
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2 What is the corresponding part to ∠Q?
A ∠R B ∠K C ∠Q D ∠P J
K L R Q
P
ΔJKL ≅ ΔPQR
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2 What is the corresponding part to ∠Q?
A ∠R B ∠K C ∠Q D ∠P J
K L R Q
P
ΔJKL ≅ ΔPQR
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Ans
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B
Slide 17 (Answer) / 183
3 What is the corresponding part to QP? A JL B LK C KJ D PQ
J
K L R Q
P
ΔJKL ≅ ΔPQR
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3 What is the corresponding part to QP? A JL B LK C KJ D PQ
J
K L R Q
P
ΔJKL ≅ ΔPQR
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Ans
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C
Slide 18 (Answer) / 183
Z
X
CV
B
4 The congruence statement for the two triangles is:
A ΔBVC ≅ ΔXCZ
B ΔXCB ≅ ΔBCX
C ΔVBC ≅ ΔZXC
D ΔCBV ≅ ΔCZX
Slide 19 / 183
Z
X
CV
B
4 The congruence statement for the two triangles is:
A ΔBVC ≅ ΔXCZ
B ΔXCB ≅ ΔBCX
C ΔVBC ≅ ΔZXC
D ΔCBV ≅ ΔCZX
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Ans
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C
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5 Complete the congruence statement: ΔXYZ ≅ A ΔXWZ B ΔZWX C ΔWXZ D ΔZXW
Y
ZW
X
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5 Complete the congruence statement: ΔXYZ ≅ A ΔXWZ B ΔZWX C ΔWXZ D ΔZXW
Y
ZW
X
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Ans
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B
Slide 20 (Answer) / 183
Properties of Congruence and Equality
We will be using the three properties of congruence we learned earlier
Reflexive Property of CongruenceSymmetric Property of CongruenceTransitive Property of Congruence
As well as the four properties of equality we learned earlier
Reflexive Property of EqualitySymmetric Property of EqualityTransitive Property of Equality
Substitution Property of Equality
Slide 21 / 183
Proving Congruence
SSS (Side-Side-Side)
Return to Tableof Contents
Slide 22 / 183
Congruent triangles have all congruent sides and angles.
However, congruence can be proven by showing less than that.
We will prove some theorems which you can then use as shortcuts to proving two triangles congruent. It is not necessary to prove that all the angles and sides are congruent.
Proving Congruence
Slide 23 / 183
Side-Side-Side Triangle CongruenceIf two triangles have the two sides equal to two sides respectively, and also have the base equal to the base, then they also have the angles equal which are contained by the equal straight lines.
Euclid - Book 1: Proposition 8
Slide 24 / 183
Side-Side-Side Triangle CongruenceEuclid showed that:
Having three equal sides requires having three equal angles.
Therefore, having three pairs of equal sides verifies that two triangles are congruent since all their corresponding sides and angles must be congruent.
Slide 25 / 183
Side-Side-Side Triangle CongruenceEuclid's argument of this (and for some of the following postulates/theorems) was based on transposing one triangle on top of the other.
He confirmed that if all the corresponding sides are equal, once you place one triangle atop the other in the correct orientation, all the sides have to line up and all the angles must as well.
Recall the Fourth Axiom: Things which coincide with one another are equal to one another.
Slide 26 / 183
Side-Side-Side Triangle Congruence
Click here to go to the lab titled, "Triangle Congruence SSS"
This is shown below.
Can you imagine a way that the corresponding angles could be of different measure without changing the length of one of the sides?
This is often called the SSS Triangle Congruence for short.
Slide 27 / 183
Side-Side-Side Triangle Congruence
Click here to go to the lab titled, "Triangle Congruence SSS"
This is shown below.
Can you imagine a way that the corresponding angles could be of different measure without changing the length of one of the sides?
This is often called the SSS Triangle Congruence for short.
[This object is a pull tab]
Mat
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eThe Lab - Triangle Congruence SSS addresses MP3, MP4, MP5, MP6, MP7 & MP8
Slide 27 (Answer) / 183
Example 1
A F
K
BGSolution:
The congruence marks on the sides show that each of the three sides in one triangle is congruent with that of the other.
By SSS, this proves congruence.
Please note that this requires that all three sides are congruent.
Prove that ΔAFK is congruent to ΔBGK
Slide 28 / 183
Example 1
A F
K
BGSolution:
The congruence marks on the sides show that each of the three sides in one triangle is congruent with that of the other.
By SSS, this proves congruence.
Please note that this requires that all three sides are congruent.
Prove that ΔAFK is congruent to ΔBGK
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This example addresses MP3 & MP6
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F
G HK
J
Example 2
Given: FG = JK, FH = JH, and H is the midpoint of GK
Prove: ΔFGH ≅ ΔJKH
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F
G HK
J
Example 2
Given: FG = JK, FH = JH, and H is the midpoint of GK
Prove: ΔFGH ≅ ΔJKH
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Mat
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This example addresses MP3 & MP6
Additional Q's to address MP Standards:What information are we given? (MP1)What do we need to prove? (MP1)If segments have equal length, what else can we say about them? Why? (MP6 & MP3)What does the midpoint mean? (MP6)How is midpoint related to congruent segments? (MP7)
Slide 29 (Answer) / 183
Statement Reason
1 FG = JK, FH = JH and H is the midpoint of GK Given
2 FG ≅ JK, FH ≅ JHDefinition of congruent segments
3 GH ≅ HK Definition of midpoint
4 ΔFGH ≅ ΔJKH Side-Side-Side Triangle Congruence
G
F
H
J
K
Example 2
Slide 30 / 183
A
B
C H
J
K
6 ΔABC ≅ ΔHJK
TrueFalse
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A
B
C H
J
K
6 ΔABC ≅ ΔHJK
TrueFalse
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Ans
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True
Slide 31 (Answer) / 183
A
B
C H
J
K
7 ΔCAB ≅ ΔHJK
TrueFalse
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A
B
C H
J
K
7 ΔCAB ≅ ΔHJK
TrueFalse
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Ans
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False
The order of the letters in the triangle name makes a difference.
Slide 32 (Answer) / 183
R
S
T U
8 ΔSRT ≅ ΔSUT TrueFalse
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R
S
T U
8 ΔSRT ≅ ΔSUT TrueFalse
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Ans
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SSS Triangle congruence applies because ST ≅ ST ; Reflexive Property
Slide 33 (Answer) / 183
9 Provide the reason for the second step.
Statement Reason
1 RS ≅ US, RT ≅ UT Given
2 ST ≅ ST ?
3 ΔSRT ≅ ΔSUT Side-Side-Side Triangle Congruence
R
S
T U
A Given B Side-Side-Side Triangle CongruenceC Reflexive property of congruence D Substitution property of congruence E Transitive property of congruence
Ans
wer
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10 ΔABC is congruent to
A ΔQRS
B ΔSRQ
C ΔACB
D ΔRSQ
A
B C Q R
S
3
4
5 3
4
5
Slide 35 / 183
10 ΔABC is congruent to
A ΔQRS
B ΔSRQ
C ΔACB
D ΔRSQ
A
B C Q R
S
3
4
5 3
4
5
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Ans
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B
Slide 35 (Answer) / 183
Proving Congruence
SAS (Side-Angle-Side)
Return to Tableof Contents
Slide 36 / 183
Side-Angle-Side Triangle Congruence
If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend.
- Euclid's Elements - Book One: Proposition 4
Slide 37 / 183
Side-Angle-Side Triangle Congruence
As in Side-Side-Side Triangle Congruence, Euclid verifies Side-Angle-Side Triangle Congruence by superposition (transposing one triangle atop the other.) He thereby indicates that if two sides of two triangles, and the angles contained by those sides, are equal, then all of the sides and angles must be equal...showing congruence.
Slide 38 / 183
Given that two triangles have two equal corresponding sides and equal angles contained by those two equal sides, the third sides must also be equal.
Tap below and the third side of each triangle will become visible.
Side-Angle-Side Triangle Congruence
Tap to reveal third side of the triangles
Slide 39 / 183
It is clear that the third side of each triangle is completely defined by the two other sides and their included angle.
So, the third sides must also be congruent.
This is often called the SAS Triangle Congruence for short.
Side-Angle-Side Triangle Congruence
Slide 40 / 183
So, if you can show that two triangles have two sides as well as the included angle (the angle formed by the two equal sides) to be equal, then all the sides and angles are congruent and the triangles are congruent.
Side-Angle-Side Triangle Congruence
Click here to go to the lab titled, "Triangle Congruence SAS"
Slide 41 / 183
So, if you can show that two triangles have two sides as well as the included angle (the angle formed by the two equal sides) to be equal, then all the sides and angles are congruent and the triangles are congruent.
Side-Angle-Side Triangle Congruence
Click here to go to the lab titled, "Triangle Congruence SAS"
[This object is a pull tab]
Mat
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eThe Lab - Triangle Congruence SAS addresses MP3, MP4, MP5, MP6, MP7 & MP8
Slide 41 (Answer) / 183
1 2
L
M
P
N
O
Example
Given: MP ≅ NP and LP ≅ OP
Prove: ΔMLP ≅ ΔNOP
Slide 42 / 183
1 2
L
M
P
N
O
Example
Given: MP ≅ NP and LP ≅ OP
Prove: ΔMLP ≅ ΔNOP
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Mat
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This example (slides #42-44) addresses MP3 & MP6
Additional Q's to address MP Standards:What information are we given? (MP1)What do we need to prove? (MP1)How are these angles related? Why? (MP6 & MP3)
Slide 42 (Answer) / 183
11 Provide the reason for line 2.
1 2
L
M
P
N
O
Given: MP ≅ NP and LP ≅ OPProve: ΔMLP ≅ ΔNOP
Statement Reason
1 MP ≅ NP and LP ≅ OP Given
2 ∠1 ≅ ∠2 ?
3 ΔMLP ≅ ΔNOP ?
A Given B Side-Side-Side Triangle CongruenceC Side-Angle-Side Triangle CongruenceD Vertical angles are congruent E Alternate interior angles are congruent
Ans
wer
Slide 43 / 183
12 Provide the reason for line 3.
Statement Reason
1 MP ≅ NP and LP ≅ OP Given
2 ∠1 ≅ ∠2 ?
3 ΔMLP ≅ ΔNOP ?
A Given B Side-Side-Side Triangle CongruenceC Side-Angle-Side Triangle CongruenceD Vertical angles are congruent E Alternate interior angles are congruent
Given: MP ≅ NP and LP ≅ OPProve: ΔMLP ≅ ΔNOP
1 2
L
M
P
N
O
Ans
wer
Slide 44 / 183
13 What is the included angle of the given sides of the triangle?
A ∠J B ∠KC ∠L
Hint: Draw the triangle!
ΔJKL, sides KL and JK
Slide 45 / 183
13 What is the included angle of the given sides of the triangle?
A ∠J B ∠KC ∠L
Hint: Draw the triangle!
ΔJKL, sides KL and JK
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Ans
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B
Slide 45 (Answer) / 183
P
QR TV4 4
5 5
100° 100°
S
14 List the congruent parts of the triangles below. Is ΔPQR ≅ ΔSTV?
YesNo
Slide 46 / 183
P
QR TV4 4
5 5
100° 100°
S
14 List the congruent parts of the triangles below. Is ΔPQR ≅ ΔSTV?
YesNo
[This object is a pull tab]
Ans
wer
Yes, by SAS Triangle Congruence
PQ ≅ ST
∠Q ≅ ∠T
RQ ≅ VT
Slide 46 (Answer) / 183
F
GH
X
Y Z46° 46°
1010
77
Why?
15 Is ΔFGH ≅ ΔXYZ by Side-Angle-Side Triangle Congruence?
YesNo
Slide 47 / 183
F
GH
X
Y Z46° 46°
1010
77
Why?
15 Is ΔFGH ≅ ΔXYZ by Side-Angle-Side Triangle Congruence?
YesNo
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Ans
wer
No, the angles marked congruent are not the included angles.
FG ≅ XY
∠H ≅ ∠Z
HF ≅ ZX
Slide 47 (Answer) / 183
A B
C D
16 Using SAS Triangle Congruence, what information is needed to show ΔABC ≅ ΔDCB ?
A ∠DBC ≅ ∠ABCB ∠B ≅ ∠CC ∠ABD ≅ ∠DCAD ∠ABC ≅ ∠DCB
Slide 48 / 183
A B
C D
16 Using SAS Triangle Congruence, what information is needed to show ΔABC ≅ ΔDCB ?
A ∠DBC ≅ ∠ABCB ∠B ≅ ∠CC ∠ABD ≅ ∠DCAD ∠ABC ≅ ∠DCB
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Ans
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D
Slide 48 (Answer) / 183
17 What type of congruence exists between the two triangles?
A SSS Triangle CongruenceB SAS Triangle CongruenceC Not enough information
Slide 49 / 183
17 What type of congruence exists between the two triangles?
A SSS Triangle CongruenceB SAS Triangle CongruenceC Not enough information
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B
Slide 49 (Answer) / 183
18 What type of congruence exists between the two triangles?A SSS Triangle CongruenceB SAS Triangle CongruenceC Not enough information
Slide 50 / 183
18 What type of congruence exists between the two triangles?A SSS Triangle CongruenceB SAS Triangle CongruenceC Not enough information
[This object is a pull tab]
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B
Slide 50 (Answer) / 183
19 What type of congruence exists between the two triangles?
A SSS Triangle CongruenceB SAS Triangle CongruenceC Not enough information
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19 What type of congruence exists between the two triangles?
A SSS Triangle CongruenceB SAS Triangle CongruenceC Not enough information
[This object is a pull tab]
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A
Slide 51 (Answer) / 183
20 What type of congruence exists between the two triangles?
A SSS Triangle CongruenceB SAS Triangle CongruenceC Not enough information
Slide 52 / 183
20 What type of congruence exists between the two triangles?
A SSS Triangle CongruenceB SAS Triangle CongruenceC Not enough information
[This object is a pull tab]
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C
Slide 52 (Answer) / 183
21 What type of congruence exists between the two triangles?
A SSS Triangle CongruenceB SAS Triangle CongruenceC Not enough information
Slide 53 / 183
21 What type of congruence exists between the two triangles?
A SSS Triangle CongruenceB SAS Triangle CongruenceC Not enough information
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C
Slide 53 (Answer) / 183
22 What type of congruence exists between the two triangles?
A SSS Triangle CongruenceB SAS Triangle CongruenceC Not enough information
45° 45°12 12
A
B
CD
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22 What type of congruence exists between the two triangles?
A SSS Triangle CongruenceB SAS Triangle CongruenceC Not enough information
45° 45°12 12
A
B
CD
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B
Slide 54 (Answer) / 183
Return to Tableof Contents
Proving Congruence
ASA (Angle-Side-Angle)
Slide 55 / 183
Angle-Side-Angle Triangle CongruenceAnother way to prove two triangles are congruent makes use of Euclid's Fifth Postulate.
This illustration should look familiar from the unit on parallel lines. It shows non-parallel lines intersected by a transversal.
1 2
3 4
Slide 56 / 183
We know from Euclid's Fifth Postulate, that the non-parallel lines will intersect on the side of the transversal on which the sum of the interior angles is less than 180º.
In this case, that's the side of the transversal with angles 2 and 4.
1 2
3 4
Angle-Side-Angle Triangle Congruence
Slide 57 / 183
By extending the lines and decreasing angles 2 and 4, we can see where the non-parallel lines intersect. This forms a triangle in which the transversal is one side and the two non-parallel lines form the other two sides.
1 2
3 4
Angle-Side-Angle Triangle Congruence
Slide 58 / 183
You can see that we have formed a triangle on the right side of the transversal, with the transversal providing one side and the two non-parallel lines the other two sides.
Let's examine that triangle.
1 2
3 4
5
A
C
B
Angle-Side-Angle Triangle Congruence
Slide 59 / 183
We can see that ∠2 and ∠4 lead to only one possible value for ∠5, since the angles must add to 180º.
That means that two triangles with two corresponding angles which are congruent, must have their third angles equal, so they are similar...but we knew that from earlier.
1 2
3 4
5
A
C
B
Angle-Side-Angle Triangle Congruence
Slide 60 / 183
Now, we also know they have corresponding sides between the two given angles, which are congruent. That means they are not only the same shape, but also the same size.The two triangles must be congruent. This is called ASA Triangle Congruence for short.
1 2
3 4
5
A
C
B
Angle-Side-Angle Triangle Congruence
To see a visual representation of what we discussed for ASA Triangle Congruence, click the link below.
http://www.mathopenref.com/congruentasa.html
Click here to go to the lab titled, "Triangle Congruence ASA"
Slide 61 / 183
Now, we also know they have corresponding sides between the two given angles, which are congruent. That means they are not only the same shape, but also the same size.The two triangles must be congruent. This is called ASA Triangle Congruence for short.
1 2
3 4
5
A
C
B
Angle-Side-Angle Triangle Congruence
To see a visual representation of what we discussed for ASA Triangle Congruence, click the link below.
http://www.mathopenref.com/congruentasa.html
Click here to go to the lab titled, "Triangle Congruence ASA"
[This object is a pull tab]
Mat
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The Lab - Triangle Congruence ASA addresses MP3, MP4, MP5, MP6, MP7 & MP8
Slide 61 (Answer) / 183
W
X
Y
23 What is the included side between ∠X and ∠W?
A YX B YW C XW
Slide 62 / 183
W
X
Y
23 What is the included side between ∠X and ∠W?
A YX B YW C XW
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Ans
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C
Slide 62 (Answer) / 183
W
X
Y
24 What is the included side between ∠X and ∠Y ?
A XW B YX
C YW
Slide 63 / 183
W
X
Y
24 What is the included side between ∠X and ∠Y ?
A XW B YX
C YW
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Ans
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B
Slide 63 (Answer) / 183
M
N
O
P
25 What information is needed to have ASA Triangle Congruence between the two triangles?
ABCD
Slide 64 / 183
M
N
O
P
25 What information is needed to have ASA Triangle Congruence between the two triangles?
ABCD
[This object is a pull tab]
Ans
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C
NP = PN is true by the Reflexive Property. If you mark the diagram to show the corresponding congruent parts, you can see NP = PN is needed
~
~
Slide 64 (Answer) / 183
A
C
D B
26 What information is needed to have ASA Triangle Congruence between the two triangles?
ABCD
Slide 65 / 183
A
C
D B
26 What information is needed to have ASA Triangle Congruence between the two triangles?
ABCD
[This object is a pull tab]
Ans
wer
B
is true by the Reflexive Property.
Slide 65 (Answer) / 183
E
F GM
H
27 Why is ∠FME ≅ ∠GMH?
A ASA Triangle CongruenceB vertical anglesC included anglesD congruent
Slide 66 / 183
E
F GM
H
27 Why is ∠FME ≅ ∠GMH?
A ASA Triangle CongruenceB vertical anglesC included anglesD congruent
[This object is a pull tab]
Ans
wer
B
Mark the congruent vertical angles when you see two lines intersect
Slide 66 (Answer) / 183
28 What type of congruence exists between the two triangles?
A SSS Triangle CongruenceB SAS Triangle CongruenceC ASA Triangle CongruenceD Not enough information
Q R
U T S
Slide 67 / 183
28 What type of congruence exists between the two triangles?
A SSS Triangle CongruenceB SAS Triangle CongruenceC ASA Triangle CongruenceD Not enough information
Q R
U T S
[This object is a pull tab]
Ans
wer A
Slide 67 (Answer) / 183
When you have overlapping figures that share sides and/or angles, marking the diagram with the given information and separating the triangles (when needed) make it easier to understand the problem.
Another strategy that you could use is to look for repeating letters once you separate the two triangles. When 2 letters repeat, then you have a common side shared. When 1 letter repeats, then you have a common angle shared.
Strategy to Prove Congruence
Slide 68 / 183
29 What type of congruence exists between ΔJLM and ΔNLK?
A SSS Triangle CongruenceB SAS Triangle CongruenceC ASA Triangle CongruenceD Not enough information
J
M
N
K
L
Pull the triangles apart!Mark the congruent parts!Are there any common sides/angles (look for letters that repeat)?
Hints:click to reveal
click to reveal
click to reveal
Slide 69 / 183
29 What type of congruence exists between ΔJLM and ΔNLK?
A SSS Triangle CongruenceB SAS Triangle CongruenceC ASA Triangle CongruenceD Not enough information
J
M
N
K
L
Pull the triangles apart!Mark the congruent parts!Are there any common sides/angles (look for letters that repeat)?
Hints:click to reveal
click to reveal
click to reveal
[This object is a pull tab]
Ans
wer B
Slide 69 (Answer) / 183
A
B
C
Q R
30 What type of congruence exists between ΔABQ and ΔCBR?
A SSS Triangle CongruenceB SAS Triangle CongruenceC ASA Triangle CongruenceD Not enough information
Mark the diagram with the given information. Be careful you don't always use all information
Hint
click to reveal
Slide 70 / 183
A
B
C
Q R
30 What type of congruence exists between ΔABQ and ΔCBR?
A SSS Triangle CongruenceB SAS Triangle CongruenceC ASA Triangle CongruenceD Not enough information
Mark the diagram with the given information. Be careful you don't always use all information
Hint
click to reveal
[This object is a pull tab]
Ans
wer C
Slide 70 (Answer) / 183
C
B
Q R
A
B
R
31 What type of congruence exists between ΔQAR and ΔRCQ?
A SSS Triangle CongruenceB SAS Triangle CongruenceC ASA Triangle CongruenceD Not enough information
Pull the triangles apart!Mark the congruent parts!Are there any common sides/angles (look for letters that repeat)?
Hints:click to reveal
click to reveal
click to reveal
Slide 71 / 183
C
B
Q R
A
B
R
31 What type of congruence exists between ΔQAR and ΔRCQ?
A SSS Triangle CongruenceB SAS Triangle CongruenceC ASA Triangle CongruenceD Not enough information
Pull the triangles apart!Mark the congruent parts!Are there any common sides/angles (look for letters that repeat)?
Hints:click to reveal
click to reveal
click to reveal
[This object is a pull tab]
Ans
wer
A
Slide 71 (Answer) / 183
ST
ND
A
vertical
32 What type of congruence exists between ΔSAN and ΔDAT?
A SSS Triangle Congruence
B SAS Triangle Congruence
C ASA Triangle Congruence
D Not enough information
At the intersection of two lines you always have _____ angles.HintClick to Reveal Click
Given: SA ≅ DA AN ≅ AT
Slide 72 / 183
ST
ND
A
vertical
32 What type of congruence exists between ΔSAN and ΔDAT?
A SSS Triangle Congruence
B SAS Triangle Congruence
C ASA Triangle Congruence
D Not enough information
At the intersection of two lines you always have _____ angles.HintClick to Reveal Click
Given: SA ≅ DA AN ≅ AT
[This object is a pull tab]
Ans
wer B
Slide 72 (Answer) / 183
33 What type of congruence exists between the two triangles?A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D Not enough
information
Slide 73 / 183
33 What type of congruence exists between the two triangles?A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D Not enough
information
[This object is a pull tab]
Ans
wer
D
There are two Sides and one Angle congruent, but they are not in the correct order.
Slide 73 (Answer) / 183
34 What type of congruence exists between the two triangles?
A SSS Triangle Congruence
B SAS Triangle Congruence
C ASA Triangle Congruence
D Not enough information C
P M
S
A
Hint:Mark the given information into your diagram. Identifying vertical angles plays an important part. click to reveal
Given: PA ≅ MA ∠P ≅ ∠M
Slide 74 / 183
34 What type of congruence exists between the two triangles?
A SSS Triangle Congruence
B SAS Triangle Congruence
C ASA Triangle Congruence
D Not enough information C
P M
S
A
Hint:Mark the given information into your diagram. Identifying vertical angles plays an important part. click to reveal
Given: PA ≅ MA ∠P ≅ ∠M
[This object is a pull tab]
Ans
wer
C
Slide 74 (Answer) / 183
Return to Tableof Contents
Proving Congruence
AAS (Angle-Angle-Side)
Slide 75 / 183
Based on that same logic, if ANY two corresponding angles and one corresponding side of a pair of triangles are congruent, the triangles must also be congruent.
This follows from the fact that the Triangle Sum Theorem tells us that once we know the measures of two angles, we know the measure of the third, since they must add to 180º.
1 2
3 4
5
A
C
B
Angle-Angle-Side Triangle Congruence
Slide 76 / 183
Another way of looking at these two theorems is that once you show that two corresponding angles in two triangles are congruent, you know that the third angles are congruent and that the two triangles are similar. That means that they have the same shape.
If you can show a side in one of those triangles is congruent to the corresponding side of the other, you know that they are same size. Thus the scale factor, k, is 1.
If they are the same size and shape, they are congruent.
ASA and AAS Triangle Congruence
Slide 77 / 183
It is really just a formality whether you use the term ASA or AAS, since all three angles must be congruent.
However, to note the difference, if the angles are both adjacent to the side which has shown to be congruent, the reason for congruence is ASA (∆ABC ≅ ∆DEF).
If not, it is AAS (∆GHI ≅ ∆JKL).
ASA and AAS Triangle Congruence
A
B C
D
E F
G
H I
J
K L
VS.
ASA AAS
Slide 78 / 183
Example
C
A
H
T
Given: ∠H ≅ ∠C ∠HTA ≅ ∠CTA
Is ΔCTA ≅ ΔHTA ?
Slide 79 / 183
Example
C
A
H
T
Given: ∠H ≅ ∠C ∠HTA ≅ ∠CTA
Is ΔCTA ≅ ΔHTA ?
[This object is a pull tab]
Mat
h Pr
actic
eThis example (and the next two slides) addresses MP3 & MP6
Additional Q's to address MP Standards:What information are we given? (MP1)What do we need to prove? (MP1)What properties/theorems/postulates might help you? (MP1)How are the triangles related? Why? (MP6 & MP3)- Ans: looking for "adjacent"; sharing the common side AT, for reflexive property
Slide 79 (Answer) / 183
Example
1) Mark the diagram:
C
A
H
T
Given: ∠H ≅ ∠C ∠HTA ≅ ∠CTA
Slide 80 / 183
Example
2) By the reflexive property:
Therefore, ΔHTA ≅ ΔCTA by AAS Triangle Congruence. AT ≅ AT
C
A
H
T
congruence statement?
Slide 81 / 183
Given:
35 What type of congruence exists, if any, between the two triangles?
A SSS Triangle Congruence
B SAS Triangle Congruence
C ASA Triangle Congruence
D AAS Triangle Congruence
E Not enough information
D
E
F
H G
Slide 82 / 183
Given:
35 What type of congruence exists, if any, between the two triangles?
A SSS Triangle Congruence
B SAS Triangle Congruence
C ASA Triangle Congruence
D AAS Triangle Congruence
E Not enough information
D
E
F
H G
[This object is a pull tab]
Ans
wer
D What is the congruence statement?
Slide 82 (Answer) / 183
36 What type of congruence exists, if any, between the two triangles?
A SSS Triangle CongruenceB SAS Triangle CongruenceC ASA Triangle CongruenceD AAS Triangle CongruenceE Not enough information
A
B C Q
RS
Slide 83 / 183
36 What type of congruence exists, if any, between the two triangles?
A SSS Triangle CongruenceB SAS Triangle CongruenceC ASA Triangle CongruenceD AAS Triangle CongruenceE Not enough information
A
B C Q
RS
[This object is a pull tab]
Ans
wer E
Two angles congruent is not enough information.
Slide 83 (Answer) / 183
37 What type of congruence exists, if any, between the two triangles?
A SSS Triangle Congruence
B SAS Triangle Congruence
C ASA Triangle Congruence
D AAS Triangle Congruence
E Not enough information
Slide 84 / 183
37 What type of congruence exists, if any, between the two triangles?
A SSS Triangle Congruence
B SAS Triangle Congruence
C ASA Triangle Congruence
D AAS Triangle Congruence
E Not enough information
[This object is a pull tab]
Ans
wer C ASA Triangle Congruence
The two triangles share a common side which is congruent via the Reflexive Property.
Slide 84 (Answer) / 183
Q
W
E
R
T
38 What type of congruence exists, if any, between the two triangles?
A SSS Triangle CongruenceB SAS Triangle CongruenceC ASA Triangle CongruenceD AAS Triangle CongruenceE Not enough information
Slide 85 / 183
Q
W
E
R
T
38 What type of congruence exists, if any, between the two triangles?
A SSS Triangle CongruenceB SAS Triangle CongruenceC ASA Triangle CongruenceD AAS Triangle CongruenceE Not enough information
[This object is a pull tab]
Ans
wer
D AAS The vertical angles are congruent. The congruent side is nonincluded so it cannot be ASA.
Slide 85 (Answer) / 183
A
S
D
F
G
H
39 What type of congruence exists, if any, between the two triangles?
A SSS Triangle Congruence
B SAS Triangle Congruence
C ASA Triangle Congruence
D AAS Triangle Congruence
E Not enough information
Slide 86 / 183
A
S
D
F
G
H
39 What type of congruence exists, if any, between the two triangles?
A SSS Triangle Congruence
B SAS Triangle Congruence
C ASA Triangle Congruence
D AAS Triangle Congruence
E Not enough information
[This object is a pull tab]
Ans
wer
B SAS Imagine you are walking around the figures, you must encounter the congruent parts in the correct order to use SAS congruence.
Slide 86 (Answer) / 183
40 What type of congruence exists, if any, between the two triangles?
A SSS Triangle Congruence
B SAS Triangle Congruence
C ASA Triangle CongruenceD AAS Triangle CongruenceE Not enough information
Slide 87 / 183
40 What type of congruence exists, if any, between the two triangles?
A SSS Triangle Congruence
B SAS Triangle Congruence
C ASA Triangle CongruenceD AAS Triangle CongruenceE Not enough information
[This object is a pull tab]
Ans
wer
D AAS Triangle Congruence Q: Are there vertical angles in this diagram?
Slide 87 (Answer) / 183
AB
C
D
41 What type of congruence exists, if any, between the two triangles?
A SSS Triangle CongruenceB SAS Triangle CongruenceC ASA Triangle CongruenceD AAS Triangle CongruenceE Not enough information
Given: BD bisects ∠ABC,∠A ≅ ∠C
Slide 88 / 183
AB
C
D
41 What type of congruence exists, if any, between the two triangles?
A SSS Triangle CongruenceB SAS Triangle CongruenceC ASA Triangle CongruenceD AAS Triangle CongruenceE Not enough information
Given: BD bisects ∠ABC,∠A ≅ ∠C
[This object is a pull tab]
Ans
wer
D
Marking the bisected angle and the common side (reflexive) shows AAS.
Slide 88 (Answer) / 183
42 What type of congruence exists, if any, between the two triangles?
A SSS Triangle CongruenceB SAS Triangle CongruenceC ASA Triangle
CongruenceD AAS Triangle
CongruenceE Not enough
information
Slide 89 / 183
42 What type of congruence exists, if any, between the two triangles?
A SSS Triangle CongruenceB SAS Triangle CongruenceC ASA Triangle
CongruenceD AAS Triangle
CongruenceE Not enough
information
[This object is a pull tab]
Ans
wer
E Not Congruent There is no AAA Congruence. AAA does make them similar (same shape), but the size may be different.
Slide 89 (Answer) / 183
HL Congruence
Return to Tableof Contents
Slide 90 / 183
The final shortcut to proving congruence is Hypotenuse Leg Triangle Congruence, or the HL Triangle Congruence for short.
This theorem states that if two right triangles have their hypotenuses and one of their legs congruent, then the triangles are congruent.
The HL Triangle Congruence can be considered a corollary of the SSS Triangle Congruence.
Hypotenuse-Leg Triangle Congruence
Slide 91 / 183
In a right triangle, the sum of the squares of the lengths of the two legs must equal the square of the length of the hypotenuse.
c2 = a2 + b2
If we are given that for two right triangles the hypotenuse and one of the legs are equal (c1 = c2 and a1 = a2), then we know that the other leg but also be equal (b1 = b2).
Thus, HL Triangle Congruence can be considered a special case, or corollary, of the Side-Side-Side Triangle Congruence.
Hypotenuse-Leg Triangle Congruence
Slide 92 / 183
c12 = a1
2 + b12 c2
2 = a22 + b2
2
Solving for b in both equations
c12 - a1
2 = b12 c2
2 - a22 = b2
2
b12 = c1
2 - a12 b2
2 = c22 - a2
2
Substituting c1 = c2 and a1 = a2
b12 = c2
2 - a22 b2
2 = c22 - a2
2
b12 = b2
2
b1 = b2
Hypotenuse-Leg Triangle Congruence
Slide 93 / 183
c12 = a1
2 + b12 c2
2 = a22 + b2
2
Solving for b in both equations
c12 - a1
2 = b12 c2
2 - a22 = b2
2
b12 = c1
2 - a12 b2
2 = c22 - a2
2
Substituting c1 = c2 and a1 = a2
b12 = c2
2 - a22 b2
2 = c22 - a2
2
b12 = b2
2
b1 = b2
Hypotenuse-Leg Triangle Congruence
[This object is a pull tab]
Mat
h Pr
actic
e
This algebraic proof addresses MP2 & MP3
Slide 93 (Answer) / 183
A B
C
R S
T
ExampleAre these two triangles congruent?
These are right triangles, so look for HL Triangle Congruence.
Slide 94 / 183
A B
C
R S
T
ExampleAre these two triangles congruent?
These are right triangles, so look for HL Triangle Congruence.
[This object is a pull tab]
Mat
h Pr
actic
e
This example (and the next slide) addresses MP3 & MP6
Additional Q's to address MP Standards:What information are we given? (MP1)What do we need to prove? (MP1)What properties/theorems/postulates might help you? (MP1)How are the triangles related? Why? (MP6 & MP3)
Slide 94 (Answer) / 183
Example
Recall that the side opposite the right angle is the hypotenuse, and the other two sides are called legs.
Hypotenuse: AC ≅ RT
Leg: CB ≅ TS
By the HL Triangle Congruence, ∆ABC ≅ ∆RST
A B
C
R S
T
Slide 95 / 183
Side-Side-Side (SSS): three sides Side-Angle-Side (SAS): two sides and the included angle
Angle-Side-Angle (ASA): two angles and the included side
Angle-Angle-Side (AAS): two angles and one non-included side
Hypotenuse-Leg (HL): hypotenuse and one leg (right triangles)
Postulates/Theorems to Prove Triangles Congruent
To use the congruence postulates/theorems, we need to know or be able to show the following congruences between two triangles:
Slide 96 / 183
Given:
Q
R S
X
Y Z
Mark the given on the diagram. Note that it is a right triangle.
43 What type of congruence exists, if any, between the two triangles?A SSS Triangle
CongruenceB SAS Triangle
CongruenceC ASA Triangle
CongruenceD AAS Triangle
CongruenceE HL Triangle CongruenceF Not enough information
HintClick to reveal
Slide 97 / 183
Given:
Q
R S
X
Y Z
Mark the given on the diagram. Note that it is a right triangle.
43 What type of congruence exists, if any, between the two triangles?A SSS Triangle
CongruenceB SAS Triangle
CongruenceC ASA Triangle
CongruenceD AAS Triangle
CongruenceE HL Triangle CongruenceF Not enough information
HintClick to reveal
[This object is a pull tab]
Ans
wer
EQ
R S
X
Y Z
Slide 97 (Answer) / 183
L
M
N O
P
Q
If they are congruent what is the congruence statement?
44 What type of congruence exists, if any, between the two triangles?A SSS Triangle
CongruenceB SAS Triangle
CongruenceC ASA Triangle
CongruenceD AAS Triangle CongruenceE HL Triangle CongruenceF Not enough information
Slide 98 / 183
L
M
N O
P
Q
If they are congruent what is the congruence statement?
44 What type of congruence exists, if any, between the two triangles?A SSS Triangle
CongruenceB SAS Triangle
CongruenceC ASA Triangle
CongruenceD AAS Triangle CongruenceE HL Triangle CongruenceF Not enough information[This object is a pull tab]
Ans
wer C
Slide 98 (Answer) / 183
A
B
CD
E
F
If they are congruent what is the congruence statement?
45 What type of congruence exists, if any, between the two triangles?A SSS Triangle
CongruenceB SAS Triangle
CongruenceC ASA Triangle
CongruenceD AAS Triangle
CongruenceE HL Triangle CongruenceF Not enough information
Slide 99 / 183
A
B
CD
E
F
If they are congruent what is the congruence statement?
45 What type of congruence exists, if any, between the two triangles?A SSS Triangle
CongruenceB SAS Triangle
CongruenceC ASA Triangle
CongruenceD AAS Triangle
CongruenceE HL Triangle CongruenceF Not enough information [This object is a pull tab]
Ans
wer F
Slide 99 (Answer) / 183
T
U
V
W
X
Y
If they are congruent what is the congruence statement?
46 What type of congruence exists, if any, between the two triangles?A SSS Triangle
CongruenceB SAS Triangle
CongruenceC ASA Triangle
CongruenceD AAS Triangle CongruenceE HL Triangle CongruenceF Not enough information
Slide 100 / 183
T
U
V
W
X
Y
If they are congruent what is the congruence statement?
46 What type of congruence exists, if any, between the two triangles?A SSS Triangle
CongruenceB SAS Triangle
CongruenceC ASA Triangle
CongruenceD AAS Triangle CongruenceE HL Triangle CongruenceF Not enough information
[This object is a pull tab]
Ans
wer
B
Slide 100 (Answer) / 183
Q
W
EY
If they are congruent what is the congruence statement?
47 What type of congruence exists, if any, between the two triangles?
A SSS Triangle CongruenceB SAS Triangle CongruenceC ASA Triangle CongruenceD AAS Triangle CongruenceE HL Triangle CongruenceF Not enough information
Slide 101 / 183
Q
W
EY
If they are congruent what is the congruence statement?
47 What type of congruence exists, if any, between the two triangles?
A SSS Triangle CongruenceB SAS Triangle CongruenceC ASA Triangle CongruenceD AAS Triangle CongruenceE HL Triangle CongruenceF Not enough information
[This object is a pull tab]
Ans
wer
D
Slide 101 (Answer) / 183
N
M
O
J
K
L
If they are congruent what is the congruence statement?
48 What type of congruence exists, if any, between the two triangles?
A SSS Triangle CongruenceB SAS Triangle CongruenceC ASA Triangle CongruenceD AAS Triangle CongruenceE HL Triangle CongruenceF Not enough information
Slide 102 / 183
N
M
O
J
K
L
If they are congruent what is the congruence statement?
48 What type of congruence exists, if any, between the two triangles?
A SSS Triangle CongruenceB SAS Triangle CongruenceC ASA Triangle CongruenceD AAS Triangle CongruenceE HL Triangle CongruenceF Not enough information
[This object is a pull tab]
Ans
wer
E
Slide 102 (Answer) / 183
E
F
G
H
If they are congruent what is the congruence statement?
49 What type of congruence exists, if any, between the two triangles?
A SSS Triangle Congruence
B SAS Triangle Congruence
C ASA Triangle Congruence
D AAS Triangle Congruence
E HL Triangle CongruenceF Not enough information
Slide 103 / 183
E
F
G
H
If they are congruent what is the congruence statement?
49 What type of congruence exists, if any, between the two triangles?
A SSS Triangle Congruence
B SAS Triangle Congruence
C ASA Triangle Congruence
D AAS Triangle Congruence
E HL Triangle CongruenceF Not enough information
[This object is a pull tab]
Ans
wer
E
Slide 103 (Answer) / 183
If they are congruent what is the congruence statement?
50 What type of congruence exists, if any, between the two triangles?
A SSS Triangle Congruence
B SAS Triangle Congruence
C ASA Triangle Congruence
D AAS Triangle Congruence
E HL Triangle CongruenceF Not enough information
E
F
G
H
Slide 104 / 183
If they are congruent what is the congruence statement?
50 What type of congruence exists, if any, between the two triangles?
A SSS Triangle Congruence
B SAS Triangle Congruence
C ASA Triangle Congruence
D AAS Triangle Congruence
E HL Triangle CongruenceF Not enough information
E
F
G
H
[This object is a pull tab]
Ans
wer
B
Slide 104 (Answer) / 183
If they are congruent what is the congruence statement?
51 What type of congruence exists, if any, between the two triangles?
A SSS Triangle CongruenceB SAS Triangle CongruenceC ASA Triangle CongruenceD AAS Triangle CongruenceE HL Triangle CongruenceF Not enough information
K F
B M
Slide 105 / 183
If they are congruent what is the congruence statement?
51 What type of congruence exists, if any, between the two triangles?
A SSS Triangle CongruenceB SAS Triangle CongruenceC ASA Triangle CongruenceD AAS Triangle CongruenceE HL Triangle CongruenceF Not enough information
K F
B M
[This object is a pull tab]
Ans
wer
A
Slide 105 (Answer) / 183
P O
Y
alternate interior
If they are congruent what is the congruence statement?
U
What angles are congruent when parallel lines are cut by a transversal?
52 What type of congruence exists, if any, between the two triangles?
A SSS Triangle Congruence
B SAS Triangle Congruence
C ASA Triangle Congruence
D AAS Triangle Congruence
E HL Triangle Congruence
F Not enough information
Click to Reveal
Slide 106 / 183
P O
Y
alternate interior
If they are congruent what is the congruence statement?
U
What angles are congruent when parallel lines are cut by a transversal?
52 What type of congruence exists, if any, between the two triangles?
A SSS Triangle Congruence
B SAS Triangle Congruence
C ASA Triangle Congruence
D AAS Triangle Congruence
E HL Triangle Congruence
F Not enough information
Click to Reveal
[This object is a pull tab]
Ans
wer
D
Slide 106 (Answer) / 183
O K
MJ
If they are congruent what is the congruence statement?
53 What type of congruence exists, if any, between the two triangles?
A SSS Triangle Congruence
B SAS Triangle Congruence
C ASA Triangle Congruence
D AAS Triangle Congruence
E HL Triangle CongruenceF Not enough information
Slide 107 / 183
O K
MJ
If they are congruent what is the congruence statement?
53 What type of congruence exists, if any, between the two triangles?
A SSS Triangle Congruence
B SAS Triangle Congruence
C ASA Triangle Congruence
D AAS Triangle Congruence
E HL Triangle CongruenceF Not enough information
[This object is a pull tab]
Ans
wer F
Slide 107 (Answer) / 183
A S
XZ
If they are congruent what is the congruence statement?
54 What type of congruence exists, if any, between the two triangles?
A SSS Triangle CongruenceB SAS Triangle CongruenceC ASA Triangle CongruenceD AAS Triangle CongruenceE HL Triangle CongruenceF Not enough information
Slide 108 / 183
A S
XZ
If they are congruent what is the congruence statement?
54 What type of congruence exists, if any, between the two triangles?
A SSS Triangle CongruenceB SAS Triangle CongruenceC ASA Triangle CongruenceD AAS Triangle CongruenceE HL Triangle CongruenceF Not enough information
[This object is a pull tab]
Ans
wer
E
Slide 108 (Answer) / 183
Triangle Congruence Proofs
Return to Tableof Contents
Slide 109 / 183
First: identify given information
Second: use a diagram that is marked with given information
Third: review congruence postulates/theorems - what information is needed (sides/angles) to use one of these?
SSS SAS ASA AAS HL
Strategy to Prove Congruence
Slide 110 / 183
First: identify given information
Second: use a diagram that is marked with given information
Third: review congruence postulates/theorems - what information is needed (sides/angles) to use one of these?
SSS SAS ASA AAS HL
Strategy to Prove Congruence
[This object is a pull tab]
Mat
h Pr
actic
eThis example (this slide and the next two) addresses MP1, MP3 & MP6
Slide 110 (Answer) / 183
E
FM
G
90°
90°8
8H
Example
Given: MF = MH = 8 and m∠F = m∠H = 90º
Prove: ΔEFM ≅ ΔGHM
Slide 111 / 183
55 Provide the reason for line 3.A Angle-Side-Angle Triangle
CongruenceB Side-Side-Side Triangle
CongruenceC Side-Angle-Side Triangle
CongruenceD Vertical angles are congruent E Alternate interior angles are congruent
E
FM
G
90°
90°8
8H
Given: MF = MH = 8 and m∠F = m∠H = 90º
Prove: ΔEFM ≅ ΔGHM
Statement Reason
1 MF = MH = 8 and m∠F = m∠H = 90º Given
2 MF ≅ MH and ∠F ≅ ∠H Defn. of congruence
3 ∠FME ≅ ∠HMG ?
4 ΔEFM ≅ ΔGHM ?
Ans
wer
Slide 112 / 183
56 Provide the reason for line 4.A Angle-Side-Angle Triangle
CongruenceB Side-Side-Side Triangle
CongruenceC Side-Angle-Side Triangle
CongruenceD Vertical angles are congruent E Alternate interior angles are congruent
E
FM
G
90°
90°8
8H
Given: MF = MH = 8 and m∠F = m∠H = 90º
Prove: ΔEFM ≅ ΔGHM
Statement Reason
1 MF = MH = 8 and m∠F = m∠H = 90º Given
2 MF ≅ MH and ∠F ≅ ∠H Defn. of congruence
3 ∠FME ≅ ∠HMG ?
4 ΔEFM ≅ ΔGHM ?
Ans
wer
Slide 113 / 183
Congruent Reasons Summary
SSS
SSASAS
AASASA
AAA
HL0
3
1
2
# of congruent angles
postulate/theorem
(Drag those that don't work out of the chart. Then put HL where it would belong.)
Slide 114 / 183
Congruent Reasons Summary
SSS
SSASAS
AASASA
AAA
HL0
3
1
2
# of congruent angles
postulate/theorem
(Drag those that don't work out of the chart. Then put HL where it would belong.)
[This object is a pull tab]
Ans
wer
SSS
SAS
AASASA
0
3
1
2
# of congruent angles
postulate/theorem
SSAAAA
HL
Slide 114 (Answer) / 183
Example
A F
K
GB
Solution (two-column):
1) Given
2) SSS Triangle Congruence
AF ≅ BG, FK ≅ GK KA ≅ KB
1)
2) ΔAFK ≅ ΔBGK
Statements Reasons
Given: AF ≅ BG, FK ≅ GK, and KA ≅ KBProve: ΔAFK ≅ ΔBGK
Slide 115 / 183
Example
A F
K
GB
Solution (two-column):
1) Given
2) SSS Triangle Congruence
AF ≅ BG, FK ≅ GK KA ≅ KB
1)
2) ΔAFK ≅ ΔBGK
Statements Reasons
Given: AF ≅ BG, FK ≅ GK, and KA ≅ KBProve: ΔAFK ≅ ΔBGK
[This object is a pull tab]
Mat
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This example addresses MP3 & MP6
Slide 115 (Answer) / 183
3. Reflexive property
2. Definition of ∠bisector
4. SAS Triangle Congruence
Example
A
B
C
DStatements Reasons
1. Given
click ___________
click ___________
click ___________
Given: BC ≅ CD AC bisects ∠BCD
Prove: ∆ABC ≅ ∆ADC
1. BC ≅ CD, AC bisects ∠BCD
2. ∠BCA ≅ ∠DCA
3. AC ≅ AC
4. ∆ABC ≅ ∆ADCclick
Slide 116 / 183
3. Reflexive property
2. Definition of ∠bisector
4. SAS Triangle Congruence
Example
A
B
C
DStatements Reasons
1. Given
click ___________
click ___________
click ___________
Given: BC ≅ CD AC bisects ∠BCD
Prove: ∆ABC ≅ ∆ADC
1. BC ≅ CD, AC bisects ∠BCD
2. ∠BCA ≅ ∠DCA
3. AC ≅ AC
4. ∆ABC ≅ ∆ADCclick
[This object is a pull tab]
Mat
h Pr
actic
e
This example (this slide and the next) addresses MP3 & MP6
Additional Q's to address MP Standards:What information are we given? (MP1)What do we need to prove? (MP1)What properties/theorems/postulates might help you? (MP1)How are the triangles related? Why? (MP6 & MP3)
Slide 116 (Answer) / 183
Problem
D
FG
EWrite a two-column proof.Given: DE ‖ FG
DE ≅ FG
Prove: ∆DEG ≅ ∆FGE
Slide 117 / 183
Problem
D
FG
EWrite a two-column proof.Given: DE ‖ FG
DE ≅ FG
Prove: ∆DEG ≅ ∆FGE
[This object is a pull tab]
Mat
h Pr
actic
e
This example (this slide and the next) addresses MP3 & MP6
Additional Q's to address MP Standards:What information are we given? (MP1)What do we need to prove? (MP1)What properties/theorems/postulates might help you? (MP1)How could you use the drawing to show your thinking? (MP5)How are the triangles related? Why? (MP6 & MP3)
Slide 117 (Answer) / 183
Problem D
FG
E
Statements Reasons
1. DE ‖ FG
2. ∠DEG ≅ ∠FGE
DE ≅ FG
3. GE ≅ EG4. ΔDEG ≅ ΔFGE
1. Given
2. Alternate Interior Angles are ≅3. Reflexive Property of Congruence4. SAS Triangle Congruence
Click
Click
Click
Click
Given: DE ‖ FGDE ≅ FG
Slide 118 / 183
A
C
DT
B
Problem: complete the proof
___________
Given: ∠A and ∠D are right angles; AT ≅ DTProve: ΔATB ≅ ΔDTC
1. ∠A and ∠D are right angles
2. ∠A ≅ ∠D
3. AT ≅ DT
4. ∠ ATB ≅ ∠ DTC
5. ΔATB ≅ ΔDTC Click Click
Click
1. Given
2. right ∠'s are congruent
3. Given
4. Vertical ∠'s are congruent
5. ASA Triangle Congruence
Click
Click
Click
Statements Reasons
Click
Slide 119 / 183
A
C
DT
B
Problem: complete the proof
___________
Given: ∠A and ∠D are right angles; AT ≅ DTProve: ΔATB ≅ ΔDTC
1. ∠A and ∠D are right angles
2. ∠A ≅ ∠D
3. AT ≅ DT
4. ∠ ATB ≅ ∠ DTC
5. ΔATB ≅ ΔDTC Click Click
Click
1. Given
2. right ∠'s are congruent
3. Given
4. Vertical ∠'s are congruent
5. ASA Triangle Congruence
Click
Click
Click
Statements Reasons
Click
[This object is a pull tab]
Mat
h Pr
actic
e
This example addresses MP3 & MP6
Additional Q's to address MP Standards:What information are we given? (MP1)What do we need to prove? (MP1)What properties/theorems/postulates might help you? (MP1)How could you use the drawing to show your thinking? (MP5)How are the triangles related? Why? (MP6 & MP3)
Slide 119 (Answer) / 183
Problem: complete the proofD C
A B
Given: DA ⊥ AB
Prove: ΔDAB ≅ ΔBCD
BC ⊥ CD∠ADB ≅ ∠CBD
Statements Reasons
1. DA ⊥ AB, BC ⊥ CD
2. ∠A and ∠C are right ∠'s
3. ∠A ≅ ∠C
4. ∠ADB ≅ ∠CBD
5. DB ≅ BD
6. ΔDAB ≅ ΔBCD
1. Given
2. Definition of ⊥ lines3. All right angles are congruent4. Given
5. reflexive property of ≅6. AAS Triangle Congruence
Click Click
ClickClick
Click
Click
Click
Click
Click
Slide 120 / 183
Problem: complete the proofD C
A B
Given: DA ⊥ AB
Prove: ΔDAB ≅ ΔBCD
BC ⊥ CD∠ADB ≅ ∠CBD
Statements Reasons
1. DA ⊥ AB, BC ⊥ CD
2. ∠A and ∠C are right ∠'s
3. ∠A ≅ ∠C
4. ∠ADB ≅ ∠CBD
5. DB ≅ BD
6. ΔDAB ≅ ΔBCD
1. Given
2. Definition of ⊥ lines3. All right angles are congruent4. Given
5. reflexive property of ≅6. AAS Triangle Congruence
Click Click
ClickClick
Click
Click
Click
Click
Click[This object is a pull tab]
Mat
h Pr
actic
e
This example addresses MP3 & MP6
Additional Q's to address MP Standards:What information are we given? (MP1)What do we need to prove? (MP1)What properties/theorems/postulates might help you? (MP1)How could you use the drawing to show your thinking? (MP5)How are the triangles related? Why? (MP6 & MP3)
Slide 120 (Answer) / 183
Statements Reasons
1)
2)
3)
4)
5)
1)
2)
3)
4)
5)
Given: AC ≅ BD, E is the midpoint of AB and CD
Prove: ΔAEC ≅ ΔBED
A
B
DC
E
Problem
E is the midpoint of AB and CD
SSS Triangle Congruence
AC = BD~
Def. of midpoint
AE = BE~Given
∆AEC ≅ ∆BEDCE = DE~
Slide 121 / 183
Statements Reasons
1)
2)
3)
4)
5)
1)
2)
3)
4)
5)
Given: AC ≅ BD, E is the midpoint of AB and CD
Prove: ΔAEC ≅ ΔBED
A
B
DC
E
Problem
E is the midpoint of AB and CD
SSS Triangle Congruence
AC = BD~
Def. of midpoint
AE = BE~Given
∆AEC ≅ ∆BEDCE = DE~
Teac
her N
otes
[This object is a teacher notes pull tab]
Given1)
2)
3)
4)
5)
Statements Reasons
1)
2)
3)
4)
5) SSS Triangle Congruence
Def. of midpoint
Def. of midpoint
AC = BD~
CE = DE~
AE = BE~
~AEC = BED
E is the midpoint of AB and CD Given
This example can be solved by matching the statements & reasons with their appropriate location.
Ans. is given below.
Slide 121 (Answer) / 183
Return to Tableof Contents
CPCTCCorresponding Parts of Congruent Triangles are Congruent
Slide 122 / 183
CPCTC states that if two or more triangles are congruent by one of the congruence postulates/theorems - SSS, SAS, ASA, AAS, or HL, then all of their corresponding parts are also congruent.
Corresponding Parts of Congruent Triangles are Congruent
CPCTC
Sometimes, our goal is not to prove two triangles congruent, but to show that a pair of corresponding sides or angles are congruent, or that some other property is true.
Slide 123 / 183
Process for proving that two segments or angles are congruent
1. Find two triangles in which the two sides or two angles are corresponding parts
2. Prove that the two triangles are congruent (SSS, SAS, ASA, AAS, HL)
3. State that the two parts are congruent, using as the reason: "Corresponding Parts of Congruent Triangles are Congruent"
Slide 124 / 183
MN
O
E L
57 Which two triangles might you try to prove congruent in order to prove NM ≅ NO ?
ABCD
ΔLOEΔNOLΔLMEΔNME
Slide 125 / 183
MN
O
E L
57 Which two triangles might you try to prove congruent in order to prove NM ≅ NO ?
ABCD
ΔLOEΔNOLΔLMEΔNME
[This object is a pull tab]
Ans
wer B and D
Slide 125 (Answer) / 183
58 Which two triangles might you try to prove congruent in order to prove EO ≅ LM ?
A
BCD
ΔEOLΔNOLΔLMEΔNME
MN
O
E L
Slide 126 / 183
58 Which two triangles might you try to prove congruent in order to prove EO ≅ LM ?
A
BCD
ΔEOLΔNOLΔLMEΔNME
MN
O
E L
[This object is a pull tab]
Ans
wer A and C
Slide 126 (Answer) / 183
59 Which two triangles might you try to prove congruent in order to prove ∠1 ≅ ∠2?
ABCD
ΔLOE
ΔNOLΔLME
ΔNME
MN
O
E L
1 2
Ans
wer
Slide 127 / 183
60 Which two triangles might you try to prove congruent in order to prove EN ≅ LN ?
ABCD
ΔLOEΔNOLΔLMEΔNME
MN
O
E L
Slide 128 / 183
60 Which two triangles might you try to prove congruent in order to prove EN ≅ LN ?
ABCD
ΔLOEΔNOLΔLMEΔNME
MN
O
E L
[This object is a pull tab]
Ans
wer B and D
Slide 128 (Answer) / 183
Statements Reasons
Problem: complete the proof
Given: AB ≅ DE, BC ≅ EC, C is the midpoint of AD
Prove: ∠A ≅ ∠DA
B
CD
E
1. AB ≅ DE
2. BC ≅ EC
3. C is the midpoint of AD
4. CA ≅ CD
5. ΔABC ≅ ΔDEC
6. ∠A ≅ ∠D
1. Given
2. Given
3. Given
4. Definition of midpoint
5. SSS Triangle Congruence
6. CPCTCClick
Click
Click
Click
Click
Click
Click
Slide 129 / 183
Statements Reasons
Problem: complete the proof
Given: AB ≅ DE, BC ≅ EC, C is the midpoint of AD
Prove: ∠A ≅ ∠DA
B
CD
E
1. AB ≅ DE
2. BC ≅ EC
3. C is the midpoint of AD
4. CA ≅ CD
5. ΔABC ≅ ΔDEC
6. ∠A ≅ ∠D
1. Given
2. Given
3. Given
4. Definition of midpoint
5. SSS Triangle Congruence
6. CPCTCClick
Click
Click
Click
Click
Click
Click[This object is a pull tab]
Mat
h Pr
actic
eThis example addresses MP3 & MP6
Additional Q's to address MP Standards:What information are we given? (MP1)What do we need to prove? (MP1)What properties/theorems/postulates might help you? (MP1)How could you use the drawing to show your thinking? (MP5)How are the triangles related? Why? (MP6 & MP3)
Slide 129 (Answer) / 183
ProblemAB
C
D E
We are given that ∠BCA ≅ ∠DCE, BC ≅ CD, and ∠B and ∠D are right angles. Since all right angles are congruent, ∠B ≅ ∠D. With the congruent angles and segments, we can conclude that ΔABC ≅ ΔEDC by ASA. Therefore, BA ≅ DE by CPCTC.
Given: ∠BCA ≅ ∠DCE ∠B and ∠D are right angles BC ≅ CDProve: BA ≅ DE
Click Click
Click
Click
Click
Slide 130 / 183
ProblemAB
C
D E
We are given that ∠BCA ≅ ∠DCE, BC ≅ CD, and ∠B and ∠D are right angles. Since all right angles are congruent, ∠B ≅ ∠D. With the congruent angles and segments, we can conclude that ΔABC ≅ ΔEDC by ASA. Therefore, BA ≅ DE by CPCTC.
Given: ∠BCA ≅ ∠DCE ∠B and ∠D are right angles BC ≅ CDProve: BA ≅ DE
Click Click
Click
Click
Click
[This object is a pull tab]
Mat
h Pr
actic
e
This example addresses MP3 & MP6
Additional Q's to address MP Standards:What information are we given? (MP1)What do we need to prove? (MP1)What properties/theorems/postulates might help you? (MP1)How could you use the drawing to show your thinking? (MP5)How are the triangles related? Why? (MP6 & MP3)
Slide 130 (Answer) / 183
Statements Reasons
Problem: complete the proof W X
P
Z Y
Given: P is the midpoint of WY, P is the midpoint of XZProve: WX ‖ ZY
1. P is the midpoint of WY2. P is the midpoint of XZ3. WP ≅ YP, ZP ≅ XP4. ∠WPX ≅ ∠YPZ 5. ΔWPX ≅ ΔYPZ6. ∠Z ≅ ∠X7. WX ‖ ZY
1. Given2. Given3. Definition of midpoint4. Vertical angles are congruent5. SAS Triangle Congruence6. CPCTC7. If alt. int. angles are congruent, then lines are parallel
Click
Click
Click
Click
Click
Click
Click
Click
Click
Click
Slide 131 / 183
Statements Reasons
Problem: complete the proof W X
P
Z Y
Given: P is the midpoint of WY, P is the midpoint of XZProve: WX ‖ ZY
1. P is the midpoint of WY2. P is the midpoint of XZ3. WP ≅ YP, ZP ≅ XP4. ∠WPX ≅ ∠YPZ 5. ΔWPX ≅ ΔYPZ6. ∠Z ≅ ∠X7. WX ‖ ZY
1. Given2. Given3. Definition of midpoint4. Vertical angles are congruent5. SAS Triangle Congruence6. CPCTC7. If alt. int. angles are congruent, then lines are parallel
Click
Click
Click
Click
Click
Click
Click
Click
Click
Click
[This object is a pull tab]
Mat
h Pr
actic
e
This example addresses MP3 & MP6
Additional Q's to address MP Standards:What information are we given? (MP1)What do we need to prove? (MP1)What properties/theorems/postulates might help you? (MP1)How could you use the drawing to show your thinking? (MP5)How are the triangles related? Why? (MP6 & MP3)
Slide 131 (Answer) / 183
Additional Proof Practice
Website link: Interactive Proofs
Slide 132 / 183
Additional Proof Practice
Website link: Interactive Proofs
[This object is a pull tab]
Teac
her N
otes
URL for the website: http://feromax.com/cgi-bin/ProveIt.pl
Proofs for Congruent Triangles can be found at the bottom of the list. All require CPCTC.
They could be done as a class, or in small groups w/ the use of laptops, iPads, Chromebooks, etc.
Slide 132 (Answer) / 183
Learning mathematics is like climbing a ladder, one step leads to the next. No step is more difficult than the one before it, as long as you take them one step at a time.
Congruent Triangles are an important step in geometry. They will be used through much of the rest of this course.
For example, the following PARCC-type question looks like it's about parallelograms, but you can answer every part of this question with what you know already, before you even study quadrilaterals.
Try it out.
Using What You've Learned
Slide 133 / 183
A
B C
DGiven: ∠BAC ≅ ∠DCA, BA ≅ DC
Prove: ABCD is a parallelogram
Statements Reasons
1. ∠BAC ≅ ∠DCA, BA ≅ DC
2. AC ≅ CA
3. ΔBAC ≅ ΔDCA4. ∠BCA ≅ ∠DAC
5. BC || AD, AB || DC
6. ABCD is a parallelogram
1. Given2. Reflexive Property of Congruence3. SAS Triangle Congruence4. CPCTC5. If alt. int. angles are congruent, then lines are parallel6. Definition of a parallelogram Click
Click Click
Click
Click
Click
Slide 134 / 183
Isosceles Triangle
Theorems
Return to Tableof Contents
Slide 135 / 183
Isosceles Triangles
leg leg
base
In an isosceles triangle, the base is the side that is not necessarily congruent to the other two sides (legs).
If an isosceles triangle has 3 congruent sides, it is also an equilateral triangle.
Slide 136 / 183
base angles
vertex angle
Isosceles Triangles
The vertex angle is opposite the base, and is the included angle between the legs.
The base angles are the angles opposite the legs, and are included by a leg and the base.
Slide 137 / 183
Base Angles Theorem
The base angles of an isosceles triangle are congruent.
This says that the angles opposite equal sides of a triangle are of equal measure.
In isosceles triangles the angles at the base are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one another.
Euclid: Book One Proposition 5
Slide 138 / 183
Proof of Base Angles Theorem
Given: In ΔABC, AB ≅ BCProve: ∠A ≅ ∠C
A
B
C
There are several ways to prove this. Euclid's way is pretty complicated.
The link below shows two typical proofs and an alternate third one.
The third proof uses the fact that order DOES matter in making statements of congruence.
It was supposedly generated by a computer.
http://www.qc.edu.hk/math/Junior%20Secondary/isosceles%20triangle.htm
Slide 139 / 183
Proof of Base Angles Theorem
Given: In ΔABC, AB ≅ BCProve: ∠A ≅ ∠C
A
B
C
There are several ways to prove this. Euclid's way is pretty complicated.
The link below shows two typical proofs and an alternate third one.
The third proof uses the fact that order DOES matter in making statements of congruence.
It was supposedly generated by a computer.
http://www.qc.edu.hk/math/Junior%20Secondary/isosceles%20triangle.htm
[This object is a pull tab]
Mat
h Pr
actic
e
This example addresses MP3 & MP6
Additional Q's to address MP Standards:What information are we given? (MP1)What do we need to prove? (MP1)What properties/theorems/postulates might help you? (MP1)How could you use the drawing to show your thinking? (MP5)How are the triangles related? Why? (MP6 & MP3)
Slide 139 (Answer) / 183
Statement Reason
1 In ΔABC, AB ≅ BC Given
2 BC ≅ AB Symmetric Property of ≅3 ∠ABC ≅ ∠CBA Reflexive Property of ≅4 ΔABC ≅ ΔCBA SAS Triangle Congruence
5 ∠A ≅ ∠C CPCTC
Proof of Base Angles Theorem
Given: In ΔABC, AB ≅ BC
Prove: ∠A ≅ ∠CA
B
C
Below are the arguments that could be used to explain the third proof from the link on the previous slide (computer generated).
Slide 140 / 183
61 What is the value of x in this triangle? Justify your answer.
x°
y°
44°
Slide 141 / 183
61 What is the value of x in this triangle? Justify your answer.
x°
y°
44°[This object is a pull tab]
Ans
wer 44º
Slide 141 (Answer) / 183
62 What is the value of y in this triangle? Justify your answer.
x°
y°
44°
Slide 142 / 183
62 What is the value of y in this triangle? Justify your answer.
x°
y°
44°[This object is a pull tab]
Ans
wer 92º
Slide 142 (Answer) / 183
x°
y°
72°
63 Solve for x and y. Explain your reasoning.
Slide 143 / 183
x°
y°
72°
63 Solve for x and y. Explain your reasoning.
[This object is a pull tab]
Ans
wer x = 36
y = 72
Slide 143 (Answer) / 183
64 What is the measure of each base angle?
70°
Slide 144 / 183
64 What is the measure of each base angle?
70°
[This object is a pull tab]
Ans
wer
70 + 2x = 1802x = 110x = 55
Each base angle is 55º
70°
x° x°
Slide 144 (Answer) / 183
65 The vertex angle of an isosceles triangle is 38°. What is the measure of each base angle?
A 71° B 38° C 83° D 104°
Slide 145 / 183
65 The vertex angle of an isosceles triangle is 38°. What is the measure of each base angle?
A 71° B 38° C 83° D 104°
[This object is a pull tab]
Ans
wer
A
Slide 145 (Answer) / 183
Converse of Base Angles TheoremIf two angles of a triangle are congruent, then the
sides opposite them are congruent.
The sides opposite equal angles of a triangle are of equal length.
If in a triangle two angles be equal to one another, the sides opposite those angles will also be equal to one another
Euclid: Book One Proposition 6
Slide 146 / 183
D
EF
4
3
66 What is the length of FD?
Slide 147 / 183
D
EF
4
3
66 What is the length of FD?
[This object is a pull tab]
Ans
wer
FD = 4
Slide 147 (Answer) / 183
D
EF
9
7
67 What is the length of ED?
Slide 148 / 183
D
EF
9
7
67 What is the length of ED?
[This object is a pull tab]
Ans
wer
ED = 9
Slide 148 (Answer) / 183
Equilateral TrianglesIf three sides of a triangle are equal, each of the
three angles has a measure of 60º .
An equilateral triangle is a special case of an isosceles triangle.
The base and legs are all of equal length.
Since all the angles are opposite sides of equal length, they all have equal measure.
Since the three angles add to 180º and have equal measure, they each have a measure of 60º.
Conversely, if two angles of a triangle are each 60º, the third angle also has a measure of 60º and all the sides are of
equal length.
Slide 149 / 183
Equilateral TrianglesIf three sides of a triangle are equal, each of the
three angles has a measure of 60º .
Conversely, if the angles of a triangle each have a measure of 60º, all the sides are of equal length.
Also, if two angles of a triangle each have a measure of 60º, the third angle must also has a measure of 60º since the Interior Angles Theorem indicates that the angles must add to 180º
Then, all the sides must be of equal length.
Slide 150 / 183
68 Classify the triangle by sides and angles.
A equilateralB isoscelesC scaleneD equiangular
E acuteF obtuseG right
7
40º
Slide 151 / 183
68 Classify the triangle by sides and angles.
A equilateralB isoscelesC scaleneD equiangular
E acuteF obtuseG right
7
40º
[This object is a pull tab]
Ans
wer
B and E
Slide 151 (Answer) / 183
69 Classify the triangle by sides and angles.
A equilateralB isoscelesC scaleneD equiangular
E acuteF obtuseG right
4
4
4
Slide 152 / 183
69 Classify the triangle by sides and angles.
A equilateralB isoscelesC scaleneD equiangular
E acuteF obtuseG right
4
4
4
[This object is a pull tab]
Ans
wer
A, B, D & E
Slide 152 (Answer) / 183
5
113º
3 3
70 Classify the triangle by sides and angles.
A equilateralB isoscelesC scaleneD equiangular
E acuteF obtuse
G right
Slide 153 / 183
5
113º
3 3
70 Classify the triangle by sides and angles.
A equilateralB isoscelesC scaleneD equiangular
E acuteF obtuse
G right
[This object is a pull tab]
Ans
wer
B and F
Slide 153 (Answer) / 183
71 Classify the triangle by sides and angles.
A equilateralB isoscelesC scaleneD equiangular
E acuteF obtuse
G right
12
12
Slide 154 / 183
71 Classify the triangle by sides and angles.
A equilateralB isoscelesC scaleneD equiangular
E acuteF obtuse
G right
12
12
[This object is a pull tab]
Ans
wer
A, B, D & E
Slide 154 (Answer) / 183
72 Classify the triangle by sides and angles.
A equilateralB isoscelesC scaleneD equiangular
E acuteF obtuse
G right
60º
60º
Slide 155 / 183
72 Classify the triangle by sides and angles.
A equilateralB isoscelesC scaleneD equiangular
E acuteF obtuse
G right
60º
60º
[This object is a pull tab]
Ans
wer
A, B, D & E
Slide 155 (Answer) / 183
Example
Find the value of x and y. Explain your reasoning.
y°
x°
Slide 156 / 183
Example
Find the value of x and y. Explain your reasoning.
y°
x°
[This object is a pull tab]
Ans
wer
x = 60º
y = 30º
Additional Q's to address MP standards:What do you know about Triangle Sum Theorem & Base Angles Theorem that you can apply to this situation? (MP7) Can you find a shortcut to solve the problem? (MP8)- Ans: the value of y will be half the value of the angle that is adjacent to the obtuse angle
Slide 156 (Answer) / 183
73 What is the value of y?
A 120°B 70°C 55°D 125°
70°
y°
Slide 157 / 183
73 What is the value of y?
A 120°B 70°C 55°D 125°
70°
y°
[This object is a pull tab]
Ans
wer
D
Slide 157 (Answer) / 183
74 What is the value of x? Justify your answer.
A 50°B 25°C 30°D 130°
50° x°
Slide 158 / 183
74 What is the value of x? Justify your answer.
A 50°B 25°C 30°D 130°
50° x°
[This object is a pull tab]
Ans
wer
B
Slide 158 (Answer) / 183
3x - 17
28
75 Solve for x in the diagram.
A 3 2/3B 14C 15D 16
Slide 159 / 183
3x - 17
28
75 Solve for x in the diagram.
A 3 2/3B 14C 15D 16
[This object is a pull tab]
Ans
wer
C
Slide 159 (Answer) / 183
PARCC Sample Test Questions
The remaining slides in this presentation contain questions from the PARCC Sample Test. After finishing this unit, you should be able to answer these questions.
Good Luck!
Return to Table of Contents
Slide 160 / 183
76 The first step of the construction is to draw an arc centered at point A that intersects both sides of the given angle. What is established by the first step?A AB ≅ BC B AB ≅ AC
C AD ≅ AC D BD ≅ CD
Use the information provided in the animation to answer the questions about the geometric construction. (note: an online video plays demonstrating the construction)
A
CB
D
Part A
Question 18/25 Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs
PARCC Released Question (EOY)
Slide 161 / 183
76 The first step of the construction is to draw an arc centered at point A that intersects both sides of the given angle. What is established by the first step?A AB ≅ BC B AB ≅ AC
C AD ≅ AC D BD ≅ CD
Use the information provided in the animation to answer the questions about the geometric construction. (note: an online video plays demonstrating the construction)
A
CB
D
Part A
Question 18/25 Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs
PARCC Released Question (EOY)
[This object is a pull tab]
Ans
wer
B
Slide 161 (Answer) / 183
77 The construction creates congruent triangles. ABD and ACD are congruent because of the
_______________ postulate/theorem.
A side-side-sideB angle-side-angle C side-angle-side D angle-angle-side
Use the information provided in the animation to answer the questions about the geometric construction. (note: an online video plays demonstrating the construction)
A
CB
D
Part BComplete the sentence with the choices given below.
Question 18/25 Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs
Slide 162 / 183
77 The construction creates congruent triangles. ABD and ACD are congruent because of the
_______________ postulate/theorem.
A side-side-sideB angle-side-angle C side-angle-side D angle-angle-side
Use the information provided in the animation to answer the questions about the geometric construction. (note: an online video plays demonstrating the construction)
A
CB
D
Part BComplete the sentence with the choices given below.
Question 18/25 Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs
[This object is a pull tab]
Ans
wer
A
A
CB
D
Slide 162 (Answer) / 183
78 It follows that AD must be the angle bisector of ∠BAC because _________________.
A ∠ACD ≅ ∠ABDB ∠BAC ≅ ∠BDCC ∠BAD ≅ ∠CADD ∠BAD ≅ ∠ABD
Use the information provided in the animation to answer the questions about the geometric construction. (note: an online video plays demonstrating the construction)
A
CB
D
Part BComplete the sentence with the choices given below.
Question 18/25 Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs
Slide 163 / 183
78 It follows that AD must be the angle bisector of ∠BAC because _________________.
A ∠ACD ≅ ∠ABDB ∠BAC ≅ ∠BDCC ∠BAD ≅ ∠CADD ∠BAD ≅ ∠ABD
Use the information provided in the animation to answer the questions about the geometric construction. (note: an online video plays demonstrating the construction)
A
CB
D
Part BComplete the sentence with the choices given below.
Question 18/25 Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs
[This object is a pull tab]
Ans
wer
C
A
CB
D
Slide 163 (Answer) / 183
Question 2/11 Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs
Marcella drew each step of a construction of an angle bisector.
Z
B
AZ Z
B
A Z
B
A
C
Z
B
AC
Step 1 Step 2 Step 3 Step 4 Step 5
Part A
Angle Z is given in Step 1. Describe the instructions for Steps 2 through 5 of the construction.
This is a great problem and draws on a lot of what we've learned.Try it in your groups.Then we'll work on it step by step together by asking questions that break the problem into pieces.
PARCC Released Question (PBA)
Slide 164 / 183
Question 2/11 Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs
Marcella drew each step of a construction of an angle bisector.
Z
B
AZ Z
B
A Z
B
A
C
Z
B
AC
Step 1 Step 2 Step 3 Step 4 Step 5
Part A
Angle Z is given in Step 1. Describe the instructions for Steps 2 through 5 of the construction.
This is a great problem and draws on a lot of what we've learned.Try it in your groups.Then we'll work on it step by step together by asking questions that break the problem into pieces.
PARCC Released Question (PBA)
[This object is a pull tab]
Ans
wer Part A: Full Credit
Slide 164 (Answer) / 183
79 Angle Z is given in Step 1. What would be the description used to get from Step 1 to Step 2?
A Construct an arc located in the interior of angle Z using a compass centered at point B with a radius length that is congruent to the radius length used to draw the arc centered at point A. Label the intersection point of the 2 interior arcs point C.
B Construct an arc located in the interior of angle Z using a compass centered at point A and a radius greater than half of angle ZBA.
C Draw a ray ZC, which is the angle bisector of angle BZA.D Construct an arc using a compass centered at point Z and any radius
length. Label the points where the arc intersects the angle A and B.
Question 2/11 Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs
Marcella drew each step of a construction of an angle bisector.
Z
B
AZ Z
B
A Z
B
A
C
Z
B
AC
Step 1 Step 2 Step 3 Step 4 Step 5
Slide 165 / 183
79 Angle Z is given in Step 1. What would be the description used to get from Step 1 to Step 2?
A Construct an arc located in the interior of angle Z using a compass centered at point B with a radius length that is congruent to the radius length used to draw the arc centered at point A. Label the intersection point of the 2 interior arcs point C.
B Construct an arc located in the interior of angle Z using a compass centered at point A and a radius greater than half of angle ZBA.
C Draw a ray ZC, which is the angle bisector of angle BZA.D Construct an arc using a compass centered at point Z and any radius
length. Label the points where the arc intersects the angle A and B.
Question 2/11 Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs
Marcella drew each step of a construction of an angle bisector.
Z
B
AZ Z
B
A Z
B
A
C
Z
B
AC
Step 1 Step 2 Step 3 Step 4 Step 5
[This object is a pull tab]
Ans
wer
D
Slide 165 (Answer) / 183
80 What would be the description used to get from Step 2 to Step 3?A Construct an arc located in the interior of angle Z using a compass
centered at point B with a radius length that is congruent to the radius length used to draw the arc centered at point A. Label the intersection point of the 2 interior arcs point C.
B Construct an arc located in the interior of angle Z using a compass centered at point A and a radius greater than half of angle ZBA.
C Draw a ray ZC, which is the angle bisector of angle BZA.D Construct an arc using a compass centered at point Z and any radius
length. Label the points where the arc intersects the angle A and B.
Question 2/11 Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs
Marcella drew each step of a construction of an angle bisector.
Z
B
AZ Z
B
A Z
B
A
C
Z
B
AC
Step 1 Step 2 Step 3 Step 4 Step 5
Slide 166 / 183
80 What would be the description used to get from Step 2 to Step 3?A Construct an arc located in the interior of angle Z using a compass
centered at point B with a radius length that is congruent to the radius length used to draw the arc centered at point A. Label the intersection point of the 2 interior arcs point C.
B Construct an arc located in the interior of angle Z using a compass centered at point A and a radius greater than half of angle ZBA.
C Draw a ray ZC, which is the angle bisector of angle BZA.D Construct an arc using a compass centered at point Z and any radius
length. Label the points where the arc intersects the angle A and B.
Question 2/11 Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs
Marcella drew each step of a construction of an angle bisector.
Z
B
AZ Z
B
A Z
B
A
C
Z
B
AC
Step 1 Step 2 Step 3 Step 4 Step 5
[This object is a pull tab]
Ans
wer
B
Slide 166 (Answer) / 183
81 What would be the description used to get from Step 3 to Step 4?A Construct an arc located in the interior of angle Z using a compass
centered at point B with a radius length that is congruent to the radius length used to draw the arc centered at point A. Label the intersection point of the 2 interior arcs point C.
B Construct an arc located in the interior of angle Z using a compass centered at point A and a radius greater than half of angle ZBA.
C Draw a ray ZC, which is the angle bisector of angle BZA.D Construct an arc using a compass centered at point Z and any radius
length. Label the points where the arc intersects the angle A and B.
Question 2/11 Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs
Marcella drew each step of a construction of an angle bisector.
Z
B
AZ Z
B
A Z
B
A
C
Z
B
AC
Step 1 Step 2 Step 3 Step 4 Step 5
Slide 167 / 183
81 What would be the description used to get from Step 3 to Step 4?A Construct an arc located in the interior of angle Z using a compass
centered at point B with a radius length that is congruent to the radius length used to draw the arc centered at point A. Label the intersection point of the 2 interior arcs point C.
B Construct an arc located in the interior of angle Z using a compass centered at point A and a radius greater than half of angle ZBA.
C Draw a ray ZC, which is the angle bisector of angle BZA.D Construct an arc using a compass centered at point Z and any radius
length. Label the points where the arc intersects the angle A and B.
Question 2/11 Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs
Marcella drew each step of a construction of an angle bisector.
Z
B
AZ Z
B
A Z
B
A
C
Z
B
AC
Step 1 Step 2 Step 3 Step 4 Step 5
[This object is a pull tab]
Ans
wer
A
Slide 167 (Answer) / 183
82 What would be the description used to get from Step 4 to Step 5?A Construct an arc located in the interior of angle Z using a
compass centered at point B with a radius length that is congruent to the radius length used to draw the arc centered at point A. Label the intersection point of the 2 interior arcs point C.
B Construct an arc located in the interior of angle Z using a compass centered at point A and a radius greater than half of angle ZBA.
C Draw a ray ZC, which is the angle bisector of angle BZA.D Construct an arc using a compass centered at point Z and any radius
length. Label the points where the arc intersects the angle A and B.
Question 2/11 Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs
Marcella drew each step of a construction of an angle bisector.
Z
B
AZ Z
B
A Z
B
A
C
Z
B
AC
Step 1 Step 2 Step 3 Step 4 Step 5
Slide 168 / 183
82 What would be the description used to get from Step 4 to Step 5?A Construct an arc located in the interior of angle Z using a
compass centered at point B with a radius length that is congruent to the radius length used to draw the arc centered at point A. Label the intersection point of the 2 interior arcs point C.
B Construct an arc located in the interior of angle Z using a compass centered at point A and a radius greater than half of angle ZBA.
C Draw a ray ZC, which is the angle bisector of angle BZA.D Construct an arc using a compass centered at point Z and any radius
length. Label the points where the arc intersects the angle A and B.
Question 2/11 Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs
Marcella drew each step of a construction of an angle bisector.
Z
B
AZ Z
B
A Z
B
A
C
Z
B
AC
Step 1 Step 2 Step 3 Step 4 Step 5
[This object is a pull tab]
Ans
wer
C
Slide 168 (Answer) / 183
Part BMarcella wants to explain why the construction produces and angle bisector. She makes a new step with line segments AB and BC added to the construction, as shown.
Using the figure, prove that ray ZC bisects angle AZB. Be sure to justify each statement of your proof.
This is a great problem and draws on a lot of what we've learned.Try it in your groups. Then we'll work on it step by step together by asking questions that break the problem into pieces.
Question 2/11 Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs
B
Z
C
A
Slide 169 / 183
83 What have we learned that will help solve this problem?A Construction of an angle bisector w/ a compass
and straightedgeB Ways to prove triangles congruentC The corresponding parts of congruent triangles
are congruent (CPCTC)D All of the above
B
Z
C
A
Slide 170 / 183
83 What have we learned that will help solve this problem?A Construction of an angle bisector w/ a compass
and straightedgeB Ways to prove triangles congruentC The corresponding parts of congruent triangles
are congruent (CPCTC)D All of the above
B
Z
C
A
[This object is a pull tab]
Ans
wer
D All of the above
Slide 170 (Answer) / 183
84 What should be the first statement in our proof?A ZA ≅ ZBB ∠BZC ≅ ∠AZCC ∆BZC ≅ ∆AZCD ZC bisects ∠AZB
B
Z
C
A
Slide 171 / 183
84 What should be the first statement in our proof?A ZA ≅ ZBB ∠BZC ≅ ∠AZCC ∆BZC ≅ ∆AZCD ZC bisects ∠AZB
B
Z
C
A
[This object is a pull tab]
Ans
wer A
Slide 171 (Answer) / 183
85 Why can we say that these two segments in step #1 are congruent?
A CPCTCB Definition of an Angle BisectorC Reflexive Property of CongruenceD Both segments were drawn with the same compass
setting, and all radii of a given circle are congruent.
B
Z
C
A
Slide 172 / 183
85 Why can we say that these two segments in step #1 are congruent?
A CPCTCB Definition of an Angle BisectorC Reflexive Property of CongruenceD Both segments were drawn with the same compass
setting, and all radii of a given circle are congruent.
B
Z
C
A
[This object is a pull tab]
Ans
wer D
Slide 172 (Answer) / 183
86 What should be the second statement in our proof?A BC ≅ ACB ∠BZC ≅ ∠AZCC ∆BZC ≅ ∆AZCD ZC bisects ∠AZB
B
Z
C
A
Slide 173 / 183
86 What should be the second statement in our proof?A BC ≅ ACB ∠BZC ≅ ∠AZCC ∆BZC ≅ ∆AZCD ZC bisects ∠AZB
B
Z
C
A
[This object is a pull tab]
Ans
wer A
Slide 173 (Answer) / 183
87 Why can we say that these two segments in step #2 are congruent?A CPCTCB Definition of an Angle BisectorC Reflexive Property of CongruenceD Both segments were drawn with the same compass
setting, and all radii of a given circle are congruent.
B
Z
C
A
Slide 174 / 183
87 Why can we say that these two segments in step #2 are congruent?A CPCTCB Definition of an Angle BisectorC Reflexive Property of CongruenceD Both segments were drawn with the same compass
setting, and all radii of a given circle are congruent.
B
Z
C
A
[This object is a pull tab]
Ans
wer D
Slide 174 (Answer) / 183
88 What should be the third statement in our proof?A ZC ≅ ZCB ∠BZC ≅ ∠AZCC ∆BZC ≅ ∆AZCD ZC bisects ∠AZB
B
Z
C
A
Slide 175 / 183
88 What should be the third statement in our proof?A ZC ≅ ZCB ∠BZC ≅ ∠AZCC ∆BZC ≅ ∆AZCD ZC bisects ∠AZB
B
Z
C
A
[This object is a pull tab]
Ans
wer A
Slide 175 (Answer) / 183
89 Why can we say that these two segments in step #3 are congruent?A CPCTCB Definition of an Angle BisectorC Reflexive Property of CongruenceD Both segments were drawn with the same compass
setting, and all radii of a given circle are congruent.
B
Z
C
A
Slide 176 / 183
89 Why can we say that these two segments in step #3 are congruent?A CPCTCB Definition of an Angle BisectorC Reflexive Property of CongruenceD Both segments were drawn with the same compass
setting, and all radii of a given circle are congruent.
B
Z
C
A
[This object is a pull tab]
Ans
wer C
Slide 176 (Answer) / 183
90 What should be the fourth statement in our proof?A ZC ≅ ZCB ∠BZC ≅ ∠AZCC ∆BZC ≅ ∆AZCD ZC bisects ∠AZB
B
Z
C
A
Slide 177 / 183
90 What should be the fourth statement in our proof?A ZC ≅ ZCB ∠BZC ≅ ∠AZCC ∆BZC ≅ ∆AZCD ZC bisects ∠AZB
B
Z
C
A
[This object is a pull tab]
Ans
wer C
Slide 177 (Answer) / 183
91 Why can we say that these two triangles in step #4 are congruent?A SSS Triangle CongruenceB SAS Triangle CongruenceC AAS Triangle CongruenceD ASA Triangle CongruenceE HL Triangle Congruence
B
Z
C
A
Slide 178 / 183
91 Why can we say that these two triangles in step #4 are congruent?A SSS Triangle CongruenceB SAS Triangle CongruenceC AAS Triangle CongruenceD ASA Triangle CongruenceE HL Triangle Congruence
B
Z
C
A
[This object is a pull tab]
Ans
wer A
Slide 178 (Answer) / 183
92 Since we know that the triangles are congruent, what should be the fifth statement in our proof?A ZC ≅ ZCB ∠BZC ≅ ∠AZCC ∆BZC ≅ ∆AZCD ZC bisects ∠AZB
B
Z
C
A
Slide 179 / 183
92 Since we know that the triangles are congruent, what should be the fifth statement in our proof?A ZC ≅ ZCB ∠BZC ≅ ∠AZCC ∆BZC ≅ ∆AZCD ZC bisects ∠AZB
B
Z
C
A
[This object is a pull tab]
Ans
wer B
Slide 179 (Answer) / 183
93 Why can we say that these two angles in step #5 are congruent?A CPCTCB Definition of an Angle BisectorC Reflexive Property of CongruenceD Both segments were drawn with the same compass
setting, and all radii of a given circle are congruent.
B
Z
C
A
Slide 180 / 183
93 Why can we say that these two angles in step #5 are congruent?A CPCTCB Definition of an Angle BisectorC Reflexive Property of CongruenceD Both segments were drawn with the same compass
setting, and all radii of a given circle are congruent.
B
Z
C
A
[This object is a pull tab]
Ans
wer A
Slide 180 (Answer) / 183
94 Since we know that the angles are congruent, what should be the sixth statement in our proof?A ZC ≅ ZCB ∠BZC ≅ ∠AZCC ∆BZC ≅ ∆AZCD ZC bisects ∠AZB
B
Z
C
A
Slide 181 / 183
94 Since we know that the angles are congruent, what should be the sixth statement in our proof?A ZC ≅ ZCB ∠BZC ≅ ∠AZCC ∆BZC ≅ ∆AZCD ZC bisects ∠AZB
B
Z
C
A
[This object is a pull tab]
Ans
wer D
Slide 181 (Answer) / 183
95 What is the final reason in our proof?A CPCTCB Definition of an Angle BisectorC Reflexive Property of CongruenceD Both segments were drawn with the same compass
setting, and all radii of a given circle are congruent.
B
Z
C
A
Slide 182 / 183
95 What is the final reason in our proof?A CPCTCB Definition of an Angle BisectorC Reflexive Property of CongruenceD Both segments were drawn with the same compass
setting, and all radii of a given circle are congruent.
B
Z
C
A
[This object is a pull tab]
Ans
wer B
Slide 182 (Answer) / 183
Statements Reasons
1) ZA ≅ ZB 1) Both segments were drawn with the same compass setting, and all radii of a given circle are congruent.
2) AC ≅ BC 2) Both segments were drawn with the same compass setting, and all radii of a given circle are congruent.
3) ZC ≅ ZC 3) Reflexive Property of ≅4) ∆AZC ≅ ∆BZC 4) SSS Triangle ≅5) ∠AZC ≅ ∠BZC 5) CPCTC6) ZC is bisects ∠AZB 6) Definition of an Angle Bisector
Given: The construction of the figure to the right
Prove: ZC bisects ∠AZB
Below is a completed version of the proof that we just wrote.
B
Z
C
A
Slide 183 / 183