Using Congruent Triangles

Chapter 4

Objective• List corresponding parts.• Prove triangles congruent (ASA,

SAS, AAS, SSS, HL)• Prove corresponding parts

congruent (CPCTC)• Examine overlapping triangles.

Key Vocabulary - Review

• Reflexive Property• Vertical Angles• Congruent Triangles• Corresponding Parts

Review: Congruence Shortcuts

**Right triangles only: hypotenuse-leg (HL)

Congruent Triangles (CPCTC)

Two triangles are congruent triangles if and only if the corresponding parts of those congruent triangles are congruent.• Correspondin

g sides are congruent

• Corresponding angles are congruent

Example: Name the Congruence Shortcut or CBD

SAS ASA

SSSSSACBD

Name the Congruence Shortcut or CBD

SAS

SAS

SAS

Reflexive Property

Vertical Angles

Vertical Angles

Reflexive Property SS

ACBD

Your Turn: Name the Congruence Shortcut or CBD

Your Turn: Name the Congruence Shortcut or CBD

Your Turn: Name the Congruence Shortcut or CBD

ExampleIndicate the additional information needed to enable us to apply the specified congruence postulate.

For ASA:

For SAS:

B

For AAS: A AC

Your Turn:Indicate the additional information needed to enable us to apply the specified congruence postulate.

For ASA:

For SAS:

For AAS:

Using Congruent Triangles: CPCTC

• If you know that two triangles are congruent, then you can use CPCTC to prove the corresponding parts in whose triangles are congruent.

*You must prove that the triangles are congruent before you can use CPCTC*

Example 1 Use Corresponding Parts

In the diagram, AB and CD bisect each other at M. Prove that A B.

Example 1 Use Corresponding Parts

Statements Reasons1. AB and CD bisect

each other at M.Given1.

2.2.

3. 3.

5. 5.

6. 6.

4.4.

The Proof Game! Here’s your chance to play the

game that is quickly becoming a favorite among America’s teenagers: The Proof Game!

Rules:1. Guys vs. Gals2. Teams must take turns filling in the statements

and reasons in the proofs to come. 3. If the statement/reason combo is correct, team

gets 1 point. Next team continues.4. If the statement/reason combo is incorrect,

team loses 1 point. Next team fixes mistake.5. Teammates cannot help the person at the

board…he/she is on their own. Cheating loses all points!!

Number OneGiven: ∠ABD = ∠CBD, ∠ADB = ∠CDBProve: AB = CB A

B

C

D

Statement Reason

Number TwoGiven: MO = RE, ME = ROProve: ∠M = ∠R

O R

EMStatement Reason

Number ThreeGiven: SP = OP, ∠SPT = ∠OPTProve: ∠S = ∠O S

P

OT

ReasonStatement

Number FourGiven: KN = LN, PN = MNProve: KP = LM K

N

L

MP

Statement Reason

Number FiveGiven: ∠C = ∠R, TY = PYProve: CT = RP C

Y

R

PT

ReasonStatement

Number SixGiven: AT = RM, AT || RMProve: ∠AMT = ∠RTMA T

RMStatement Reason

Example 2 Visualize Overlapping Triangles

SOLUTION

Sketch the triangles separately and mark any given information. Think of ∆JGH moving to the left and ∆KHG moving to the right.

1.

Sketch the overlapping triangles separately. Mark all congruent angles and sides. Then tell what theorem or postulate you can use to show ∆JGH ∆KHG.

Mark GJH HKG and JHG KGH.

Example 2 Visualize Overlapping Triangles

Look at the original diagram for shared sides, shared angles, or any other information you can conclude.

2.

Add congruence marks to GH in each triangle.

In the original diagram, GH and HG are the same side, so GH HG.

You can use the AAS Congruence Theorem to show that ∆JGH ∆KHG.

3.

Example 3 Use Overlapping Triangles

SOLUTION

Write a proof that shows AB DE.ABC DEC

AB DE

CB CE

Your Turn: Use Overlapping Triangles

Given KJ KL and J L, show NJ ML.

Redraw the triangles separately and label all congruences. Explain how to show that the triangles or corresponding parts are congruent.

Your Turn: Use Overlapping Triangles

Given SPR QRP and Q S, show ∆PQR ∆RSP.

3.

Joke Time• What happened to the man who lost the

whole left side of his body?• He is all right now.

• What did one eye say to the other eye? • Between you and me something smells.

Upcoming Schedule• Quiz on Friday…HL, proofs, CPCTC,

Isosceles Triangle Thm, overlapping triangles• Monday – vocabulary terms• Tues – Practice Day• Wednesday – Chapter 4 Test• **reminder – projects due Oct.

27!!!