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Chapter 4 ReviewProving Triangles

Congruent and Isosceles Triangles

(SSS, SAS, ASA,AAS)

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PostulatesSSS If the sides of one triangle are congruent to the sides of a

second triangle, then the triangles are congruent.

Included Angle: In a triangle, the angle formed by two sides is the included angle for the two sides.

Included Side: The side of a triangle that forms a side of two given angles.

A

B C

D

E F

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Included Angles & Sides

& .A is the included angle for AB AC

& .B is the included angle for BA BC

& .C is the included angle for CA CB

A

B C

Included Angle:

Included Side:& .AB is the included side for A B

& .BC is the included side for B C

& .AC is the included side for A C

** *

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PostulatesASA If two angles and the included side of one triangle are

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

SAS If two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent.

A

B C

D

E F

A

B C

D

E F

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Steps for Proving Triangles Congruent

1. Mark the Given.

2. Mark … Reflexive Sides / Vertical Angles

3. Choose a Method. (SSS , SAS, ASA)

4. List the Parts … in the order of the method.

5. Fill in the Reasons … why you marked the parts.

6. Is there more?

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Problem 1 Given: AB CD BC DAProve: ABC CDA

Statements Reasons

Step 1: Mark the Given Step 2: Mark reflexive sidesStep 3: Choose a Method (SSS /SAS/ASA )Step 4: List the Parts in the order of the methodStep 5: Fill in the reasonsStep 6: Is there more?

A B

D C

SSS

1. AB CD2. BC DA3. AC AC

Given

Given

Reflexive Property

SSS Postulate4. ABC CDA

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Problem 2 Step 1: Mark the Given Step 2: Mark vertical anglesStep 3: Choose a Method (SSS /SAS/ASA)Step 4: List the Parts in the order of the methodStep 5: Fill in the reasonsStep 6: Is there more?

SAS

Given

Given

Vertical Angles.

SAS Postulate

: ;

Pr :

Given AB CB EB DB

ove ABE CBD

E

C

D

AB

1. AB CB2. ABE CBD

3. EB DB4. ABE CBD

Statements Reasons

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Problem 3

Statements Reasons

Step 1: Mark the Given Step 2: Mark reflexive sidesStep 3: Choose a Method (SSS /SAS/ASA)Step 4: List the Parts in the order of the methodStep 5: Fill in the reasonsStep 6: Is there more?

ASA

Given

Given

Reflexive Postulate

ASA Postulate

: ;

Pr :

Given XWY ZWY XYW ZTW

ove WXY WZY

Z

W Y

X 1. XWY ZWY

2. WY WY3. XYW ZYW

4. WXY WZY

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PostulatesAAS If two angles and a non included side of one triangle are

congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent.

A

B C

D

E F

Lesson 4-4: AAS & HL Postulate 10

Problem 1

Statements Reasons

Step 1: Mark the Given Step 2: Mark vertical anglesStep 3: Choose a Method (SSS /SAS/ASA/AAS/ HL )Step 4: List the Parts in the order of the methodStep 5: Fill in the reasonsStep 6: Is there more?

AAS

Given

Given

Vertical Angle Thm

AAS Postulate

Given: A C BE BDProve: ABE CBD

E

C

D

AB

1. A C2. ABE CBD

3. BE BD

4. ABE CBD

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Parts of an Isosceles Triangle

An isosceles triangle is a triangle with two congruent sides.

The congruent sides are called legs and the third side is called the base.

3

Leg Leg

Base

21

1 and 2 are base angles3 is the vertex angle

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Isosceles Triangle Theorems

By the Isosceles Triangle Theorem,the third angle must also be x.Therefore, x + x + 50 = 180

2x + 50 = 1802x = 130x = 65

Example:

x

50

Find the value of x.

A

B C

, .If AB AC then B C

If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

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Isosceles Triangle TheoremsIf two angles of a triangle are congruent, then the sides opposite those angles are congruent.

Example: Find the value of x. Since two angles are congruent, the sides opposite these angles must be congruent.

3x – 7 = x + 152x = 22X = 11

A

B C

50 50

3x - 7x+15

A

B C

, .If B C then AB AC