Using Congruent Triangles

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# Using Congruent Triangles - PowerPoint PPT Presentation

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

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### 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.

• Corresponding sides are congruent
• Corresponding angles are congruent

Name the Congruence Shortcut or CBD

Vertical Angles

Reflexive Property

SAS

SAS

Reflexive Property

Vertical Angles

SSA

SAS

CBD

Example

AC

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

For ASA:

B 

For SAS:

A

For AAS:

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

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

Use Corresponding Parts

Example 1

Statements

Reasons

1.

AB and CD bisect each other at M.

2.

2.

3.

3.

4.

4.

5.

5.

6.

6.

Use Corresponding Parts

1.

Given

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:

Guys vs. Gals

Teams must take turns filling in the statements and reasons in the proofs to come.

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

If the statement/reason combo is incorrect, team loses 1 point. Next team fixes mistake.

Teammates cannot help the person at the board…he/she is on their own. Cheating loses all points!!

Number One

Given: ∠ABD = ∠CBD, ∠ADB = ∠CDB

Prove: AB = CB

B

A

C

Statement

Reason

D

Number Two

Given: MO = RE, ME = RO

Prove: ∠M = ∠R

O

R

Statement

Reason

M

E

Number Three

Given: SP = OP, ∠SPT = ∠OPT

Prove:∠S = ∠O

O

T

S

Statement

Reason

P

Number Four

Given: KN = LN, PN = MN

Prove: KP = LM

K

L

N

Statement

Reason

M

P

Number Five

Given:∠C = ∠R, TY = PY

Prove: CT = RP

C

R

Y

Statement

Reason

P

T

Number Six

Given: AT = RM, AT || RM

Prove:∠AMT = ∠RTM

A

T

Statement

Reason

M

R

Example 2

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

SOLUTION

1.

Sketch the triangles separately and mark any given information. Think of ∆JGHmoving to the left and ∆KHGmoving to the right.

MarkGJH HKG andJHG KGH.

Visualize Overlapping Triangles

Example 2

2.

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

Add congruence marks to GHin each triangle.

3.

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

Visualize Overlapping Triangles

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

Example 3

Write a proof that shows ABDE.

ABC DEC

CB CE

AB DE

Use Overlapping Triangles

SOLUTION

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

GivenKJ KLandJ L,showNJML.

Use Overlapping Triangles

3.

Given SPR QRPand Q S, show ∆PQR  ∆RSP.

Use Overlapping Triangles

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!!!