Unit 11

CCGPS Analytic Geometry

Proving Triangles Congruent

(SSS, SAS, ASA, AAS, HL)

Unit 1 : SSS, SAS, ASA 2

SSS

If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.

A

B C

D

E F

Side-Side-SideSSS

• If all 3 sides of 2 triangles are congruent, the triangles are congruent.

If AB ED,

Make sure that you write the congruency statement so that the corresponding vertices (and thus the corresponding sides) are in the same position in thecongruency statement.

∆ABC ∆EDF

BC DF, and AC EF, then the 2 triangles are congruent

not ∆ABC ∆DEF

Lesson 4-3: SSS, SAS, ASA 4

Included Angles

& .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:

** *

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

Included Angle

Lesson 4-3: SSS, SAS, ASA 5

Included Sides

A

B C

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

** *

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

Lesson 4-3: SSS, SAS, ASA6

ASAAngle Side Angle

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.

A

B C

D

E F

EA Sides AB = ED

DB

A

S

A

EDFABC

Lesson 4-3: SSS, SAS, ASA 7

SAS Side Angle Side

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.

S

S

A

Lesson 4-3: SSS, SAS, ASA 8

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?

Lesson 4-3: SSS, SAS, ASA 9

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

Given

Given

Reflexive Property

SSS Postulate4. ABC CDA

1. AB CD2. BC DA3. AC CA

Unit 1 CCGPS Analytic Geom. SSS, SAS, ASA

10

Problem 2Step 1: Mark the Given Step 2: Mark vertical angles congruentStep 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

Lesson 4-3: SSS, SAS, ASA 11

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 ZYW

ove WXY WZY

Z

W Y

X 1. XWY ZWY

2. WY WY3. XYW ZYW

4. WXY WZY

AAS Angle Angle Side (corresponding)

DEFABC

HGJDEFABC

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.

IS NOT CONGRUENT WITH EITHER OF THE OTHER 2 Congruent Angles and side DO NOT correspond..

Hypotenuse Leg HL

If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent.

A

B C

D

E F

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

Problem 2

3. AC AC2. AB AD

1. ,ABC ADCright s

Step 1: Mark the Given Step 2: Mark reflexive sidesStep 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?

HL

Given

Given

Reflexive Property

HL Postulate

Given: ABC, ADC right s AB ADProve: ABC ADC

CB D

A

4. ABC ADC

Statements Reasons