4.2 apply congruence and triangles
TRANSCRIPT
![Page 1: 4.2 apply congruence and triangles](https://reader033.vdocuments.site/reader033/viewer/2022052316/559f02801a28ab2e378b46be/html5/thumbnails/1.jpg)
4.24.2 Apply Congruence and TrianglesBell Thinger
1. When are two angles congruent?
ANSWER when they have the same measure
ANSWER 45º
2. In ∆ABC, if m A = 64º and m B = 71º, what is m C?
ANSWER Transitive Property
3. What property of angle congruence is illustrated by this statement? If A ≅ B and B ≅ C, then A ≅
C.
![Page 2: 4.2 apply congruence and triangles](https://reader033.vdocuments.site/reader033/viewer/2022052316/559f02801a28ab2e378b46be/html5/thumbnails/2.jpg)
4.2
![Page 3: 4.2 apply congruence and triangles](https://reader033.vdocuments.site/reader033/viewer/2022052316/559f02801a28ab2e378b46be/html5/thumbnails/3.jpg)
4.2Example 1
Write a congruence statement for the triangles. Identify all pairs of congruent corresponding parts.
SOLUTION
The diagram indicates that JKL ≅ TSR.
Corresponding angles ∠J ≅ ∠T, ∠K ≅ ∠S, ∠L ≅ ∠R
Corresponding sides JK ≅ TS, KL ≅ SR, LJ ≅ RT
![Page 4: 4.2 apply congruence and triangles](https://reader033.vdocuments.site/reader033/viewer/2022052316/559f02801a28ab2e378b46be/html5/thumbnails/4.jpg)
4.2Example 2
In the diagram, DEFG ≅ SPQR.
Find the value of x.a.
SOLUTION
FG = QR
12 = 2x – 4
16 = 2x
8 = x
You know that FG ≅ QR.a.
![Page 5: 4.2 apply congruence and triangles](https://reader033.vdocuments.site/reader033/viewer/2022052316/559f02801a28ab2e378b46be/html5/thumbnails/5.jpg)
4.2Example 2
In the diagram, DEFG ≅ SPQR.
SOLUTION
b. Find the value of y.
b. You know that ∠F ≅ ∠Q.
m∠F = m∠Q
68 o = (6y + x)o
68 = 6y + 8
10 = y
![Page 6: 4.2 apply congruence and triangles](https://reader033.vdocuments.site/reader033/viewer/2022052316/559f02801a28ab2e378b46be/html5/thumbnails/6.jpg)
4.2Example 3
SOLUTION
If you divide the wall into orange and blue sections along JK , will the sections of the wall be the same size and shape?Explain.
PAINTING
From the diagram, ∠A ≅ ∠C and ∠D ≅ ∠B because all right angles are congruent. Also, by the Lines Perpendicular to a Transversal Theorem, AB || DC .
![Page 7: 4.2 apply congruence and triangles](https://reader033.vdocuments.site/reader033/viewer/2022052316/559f02801a28ab2e378b46be/html5/thumbnails/7.jpg)
4.2Example 3
Then, ∠1 ≅ ∠4 and ∠2 ≅ ∠3 by the Alternate Interior Angles Theorem. So, all pairs of corresponding angles are congruent.
The diagram shows AJ ≅ CK , KD ≅ JB , and DA ≅ BC . By the Reflexive Property, JK ≅ KJ . All corresponding parts are congruent, so AJKD ≅ CKJB.
![Page 8: 4.2 apply congruence and triangles](https://reader033.vdocuments.site/reader033/viewer/2022052316/559f02801a28ab2e378b46be/html5/thumbnails/8.jpg)
4.2Guided Practice
1. Identify all pairs of congruent corresponding parts.
ANSWER
Corresponding sides: AB ≅ CD, BG ≅ DE, GH ≅ FE, HA ≅ FC
Corresponding angles: ∠A ≅ ∠C, ∠B ≅ ∠D, ∠G ≅ ∠E, ∠H ≅ ∠F.
In the diagram at the right, ABGH ≅ CDEF.
![Page 9: 4.2 apply congruence and triangles](https://reader033.vdocuments.site/reader033/viewer/2022052316/559f02801a28ab2e378b46be/html5/thumbnails/9.jpg)
4.2
3. Show that PTS ≅ RTQ.
ANSWER
All of the corresponding parts of PTS are congruent to those of RTQ by the indicated markings, the Vertical Angle Theorem and the Alternate Interior Angle theorem.
Guided Practice
![Page 10: 4.2 apply congruence and triangles](https://reader033.vdocuments.site/reader033/viewer/2022052316/559f02801a28ab2e378b46be/html5/thumbnails/10.jpg)
4.2
In the diagram at the right, ABGH ≅ CDEF.
2. Find the value of x and find m∠H.
ANSWER 25, 105°
Guided Practice
![Page 11: 4.2 apply congruence and triangles](https://reader033.vdocuments.site/reader033/viewer/2022052316/559f02801a28ab2e378b46be/html5/thumbnails/11.jpg)
4.2
![Page 12: 4.2 apply congruence and triangles](https://reader033.vdocuments.site/reader033/viewer/2022052316/559f02801a28ab2e378b46be/html5/thumbnails/12.jpg)
4.2Example 4
Find m∠BDC.
So, m∠ACD = m∠BDC = 105° by the definition of congruent angles.
ANSWER
SOLUTION
∠A ≅∠B and ∠ADC ≅∠BCD, so by the Third Angles Theorem, ∠ACD ≅∠BDC. By the Triangle Sum Theorem, m∠ACD = 180° – 45° – 30° = 105° .
![Page 13: 4.2 apply congruence and triangles](https://reader033.vdocuments.site/reader033/viewer/2022052316/559f02801a28ab2e378b46be/html5/thumbnails/13.jpg)
4.2Example 5
Plan for Proof
b. Use the Third Angles Theorem to show that ∠B ≅ ∠D.
Write a proof.
PROVE: ACD ≅ CAB
GIVEN: AD ≅ CB, DC ≅ BA, ∠ACD ≅ ∠CAB, ∠CAD ≅ ∠ACB
a. Use the Reflexive Property to show that AC ≅ AC.
![Page 14: 4.2 apply congruence and triangles](https://reader033.vdocuments.site/reader033/viewer/2022052316/559f02801a28ab2e378b46be/html5/thumbnails/14.jpg)
4.2Example 5
Plan in Action
1. Given
2. Reflexive Property of Congruence
STATEMENTS REASONS
3. Given
4. Third Angles Theorem
1. AD ≅ CB , DC ≅ BA
2. a. AC ≅ AC.
3. ∠ACD ≅ ∠CAB,∠CAD ≅ ∠ACB
4. b. ∠B ≅ ∠D
5. ACD ≅ CAB Definition of ≅ figures5.
![Page 15: 4.2 apply congruence and triangles](https://reader033.vdocuments.site/reader033/viewer/2022052316/559f02801a28ab2e378b46be/html5/thumbnails/15.jpg)
4.2
4. In the diagram, what is m∠DCN.
ANSWER 75°
Guided Practice
![Page 16: 4.2 apply congruence and triangles](https://reader033.vdocuments.site/reader033/viewer/2022052316/559f02801a28ab2e378b46be/html5/thumbnails/16.jpg)
4.2
By the definition of congruence, whatadditional information is needed toknow that NDC ≅ NSR.
5.
Guided Practice
ANSWER
DC ≅ RS and DN ≅ SN
![Page 17: 4.2 apply congruence and triangles](https://reader033.vdocuments.site/reader033/viewer/2022052316/559f02801a28ab2e378b46be/html5/thumbnails/17.jpg)
4.2
![Page 18: 4.2 apply congruence and triangles](https://reader033.vdocuments.site/reader033/viewer/2022052316/559f02801a28ab2e378b46be/html5/thumbnails/18.jpg)
4.2Exit Slip
CAANSWER
3. EDF ≅ ?
In the diagram, ABC ≅ DEF. Complete each statement.
BACANSWER
60°ANSWER
2. FD ≅ ?
m A = ?1.
![Page 19: 4.2 apply congruence and triangles](https://reader033.vdocuments.site/reader033/viewer/2022052316/559f02801a28ab2e378b46be/html5/thumbnails/19.jpg)
4.2
4. Write a congruence statement for the two small triangles. Explain your reasoning.
Exit Slip
WXZ ≅ YXZ; The diagram tell us that W ≅ Y and WZX ≅ YZX. WXZ ≅ YXZ by the Third Thm. From the diagram WX ≅ YX and WZ ≅ YZ, and XZ ≅ XZ by Refl. Prop. Of ≅.
ANSWER
![Page 20: 4.2 apply congruence and triangles](https://reader033.vdocuments.site/reader033/viewer/2022052316/559f02801a28ab2e378b46be/html5/thumbnails/20.jpg)
4.2
Homework
Pg 238-241#5, 15, 16, 19, 20