Congruent Triangles

Polygons MNOL and ZYXW are congruent

∆ABC and ∆DEF are congruentRectangles ABCD and EFGH are not congruent

∆ZXY and ∆JLP are not congruent

A

C

D

B

H

G

E

F

YZ

X

PJ

L

4-1 Congruent Figures

Objective: To recognize congruent figures and their

corresponding parts

Vocabulary/ Key Concept• Congruent polygons-

two polygons are congruent if their corresponding sides and angles are congruent

Naming Congruent Figures

Ang Legs Triangle: Construct two triangles with the following sides-1 red, 1 blue, 1 yellow

∆ABC and ∆DEF

óA

óB

óC

Warm Up: WXYZ JKLM. List 4 pairs of congruent sides and angles.

• WX JK• XY KL• YZ LM• ZW MJ

• W J• K X• Y L• Z M

Each pair of polygons are congruent. Find the measure of each numbered angle

• M1 = 110• m 2 = 120

• M3 = 90• m 4 = 135

We know:

óB óF

óA óE

Then we can conclude:

óC óD

Key Concept: If two angles in a triangle are congruent to two angles in another triangle, then the third angles are congruent.

WARNING: This is only true for ANGLES not side lengths!

How do we know if two triangles are congruent?

Concept Check!

Objective:To prove two triangles are

congruent using SSS and SAS Postulates

Key Concepts

• SSS – Side-side-side corresponding congruence.

If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent(all corresponding sides are equal)

Example 1: State if the two triangles are congruent. If they are, write a congruence statement and state how you know they are congruent.

Student Slide #1

Example 2: State if the two triangles are congruent. If they are, write a congruence statement and state how you know they are congruent.

Student Slide #2

Key Concepts

• SAS – Side-Angle-Side corresponding Congruence.

ANGLE MUST BE IN BETWEEN THE TWO SIDES (INCLUDED ANGLE)If two sides and the included angle of one triangle are

congruent to two sides and the included angle of another triangle, then the two triangles are congruent

Example 1: State if the two triangles are congruent. If they are, write a congruence statement and state how you know they are congruent.

Student Slide #3

Example 2: Can you use SAS to prove these two triangles are congruent?If no, what information would you need in order to use SAS to prove these triangles are congruent?

Student Slide #4

Determine if you can use SSS or SAS to prove two triangles are congruent. Write the congruence statement.

ABD CBD by SAS

AB CB --CONGRUENCE MARKING BD BD – REFLEXIVE PROPERTY OF CONGRUENCEABD CBD –CONGRUENCE MARKING

óB óE

If we know:

What other information must we know in order to prove

∆ABC ∆DEF using SAS?

Example:

WARM UP (will be collected): a) Name the three pairs of corresponding sidesb) Name the three pairs of corresponding anglesc) Do we have enough information to conclude that

the two triangles are congruent? Explain your reasoning.

*CORRESPONDING DOES NOT MEAN THEY ARE CONGRUENT!

WUP#1: Determine if you can use SSS or SAS to prove two triangles are congruent. Write the congruence statement.

What do you know?NP QP -- CONGRUENT MARKSNR QR -- CONGRUENT MARKSRP RP -- REFLEXIVE PROPERTY OF

PRN PRQ by SSS

WUP #2: What one piece of additional information must we know in order to prove the triangles are congruent using SAS. Explain your reasoning and then write a congruence statement.Explanation:

Statement:

Objective: To prove two triangles are congruent using ASA, AAS,

and HL Postulates

Key Concepts• ASA – Two angles and an included side.

SIDE IS IN BETWEEN THE ANGLES

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

• AAS – Two angles and a non-included side.

Key Concepts

If two angles and the non-included side of a triangle are congruent to two angles and the non-included side of another triangle, then the two triangles are congruent.

Determine if you can use ASA or AAS to prove two triangles are congruent. Write the congruence statement.

Determine if you can use ASA or AAS to prove two triangles are congruent and explain your reasoning. Then write the congruence statement.

Explain:

Determine if you can use ASA or AAS to prove two triangles are congruent and explain your reasoning. Then write the congruence statement.

Explain:

TRY ONE

Congruence that works: Congruence that does not work:

SSS

SAS

AAS

ASA

ASS

SSA

AAA

*Remember, we don’t swear in math (not even backwards). And no screaming!

What did you learn today?

• What are the five ways (one for right triangles) to prove triangles are congruent?

Example 1:Complete the 2 column proof:

Given: óBAE óEDB, óABE óDEB

Prove:

óABE óDEB

Statements Reasons

So what do we know about the parts of congruent triangles?

Corresponding Parts of Congruent Triangles are Congruent

Hence,

CPCTC*Remember, you can only use CPCTC, AFTER you have proven two triangles to be congruent!

Write a Proof

Statement 1. FJ GH JFH GHF2. HF FH3. JFH GHF4. FG JH

Reasons 1. Given2. Reflexive property of

congruence3. SAS4. CPCTC

TRY ONE: Write a Proof

Statement 1. AC CD, óBAC

óCDE2. ACB ECD 3. DEC ABC4. B E

Reasons 1. Given2. Vertical angles3. ASA4. CPCTC

Given: óBAC óCDE, AC CD Prove: óB óE

What did you learn today?

• What does CPCTC mean and when do we use it?

CPCTC Song (sung to the tune of “YMCA” by the Village People) Author of lyrics: Eagler

Young man, there's no need to feel down I said, young man, pick yourself off the ground I said, young man, 'cause there's a new thing I've found There's no need to be unhappy

Young man, there's this thing you can do I said, young man, it's so easy to prove You can use it, and I'm sure you will see Many ways to show congruency

It's fun to solve it with C-P-C-T-C It's fun to solve it with C-P-C-T-C Barely takes any time, uses only one line It's the easiest thing you'll find

It's fun to solve it with C-P-C-T-C It's fun to solve it with C-P-C-T-C If you don't have a clue, it's so simple to do Write five letters and you'll be through

Mini Lab

Similar Polygons:

Two polygons are similar if:1)Corresponding angles are congruent2)Corresponding sides are proportional

Determine whether rectangle HJKL is similar to rectangle MNPQ.

Transformations

Reflection over the x-axis

Preserves congruence

Rotation 90ô about the origin

Preserves congruence

90ô(x,y) (-y,x)Ex: D(6,3)D’(-3,6)

180ô (x,y)(-x,-y)

Translation

Preserves congruence

3 units right,4 units down

Dilation

Does not preserve congruence

Scale factor (length of image/length of original) :2/1

BUT, are they similar?

Description:

Description:

1) Which transformations preserve congruence?2) What criteria needs to be met for triangles to be similar?

Proving Triangles Similar

There are 3 postulates that we can use to prove triangles are similar…

Angle-Angle Similarity (AA ~) Postulate:

If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

Explain why the triangles are similar. Write a similarity statement.1.óR óV

2.óRSW óVSB

3.∆RSW ~ ∆VSB

1. Equal measures

2. Vertical angles are

3. AA ~ postulate

Statements Reasons

Side-Angle-Side Similarity (SAS ~) Postulate:

If an angle of one triangle is congruent to an angle of a second and the sides including the two angles are proportional, then the two triangles are similar.

Explain why the triangles are similar. Write a similarity statement.

1.óR óY

2. And and

Therefore ∆RSW ~ ∆VSB

1. Both right angles

2. SAS ~ Postulate

Statements Reasons

Side-Side-Side Similarity (SSS ~) Postulate:

If the corresponding sides of two triangles are proportional then the two triangles are similar.

Explain why the triangles are similar. Write a similarity statement.

Determine if the two triangles are similar. If they are, write a similarity statement and find DE.

Solve Proportions.

In sunlight, a cactus casts a 9-ft shadow. At the same time a person 6 ft. tall casts a 4 ft. shadow. Use similar triangles to find the height of the cactus.

Side-Splitter Theorem: If a line is parallel to one side of a triangle and intersects the other two sides then it divides those sides proportionally.

Test Taking Tip:Whenever you see a line that passes through a triangle and is parallel to one side, the Side Splitter Theorem may apply!

Triangle Angle Bisector Theorem: If a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle.

Name the three postulates that prove two triangles are similar.