Chapter 4 Triangles

4-3 Exploring Congruent Triangles Triangles that are the same size and same shape

are congruent triangles. Each triangle has six parts, three angles and three

sides. If the corresponding six parts of one triangle are

congruent to the six parts of another triangle, then the triangles are congruent.

If you slide, rotate, or flip a figure, congruence will not change. These three transformations are called congruence transformations.

Definition of Congruent Triangles (CPCTC) Two triangles are congruent if and only if their

corresponding parts are congruent. Congruence of triangles is reflexive,

symmetric, and transitive.

SSS AND SAS CONGRUENCE POSTULATES

If all six pairs of corresponding parts (sides and angles) arecongruent, then the triangles are congruent.

and thenIfSides are congruent

1. AB DE

2. BC EF

3. AC DF

Angles are congruent

4. A D

5. B E

6. C F

Triangles are congruent

ABC DEF

SSS AND SAS CONGRUENCE POSTULATES

POSTULATE

POSTULATE 19 Side - Side - Side (SSS) Congruence Postulate

Side MN QR

Side PM SQ

Side NP RS

If

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

then MNP QRSS

S

S

SSS AND SAS CONGRUENCE POSTULATES

The SSS Congruence Postulate is a shortcut for provingtwo triangles are congruent without using all six pairsof corresponding parts.

Using the SSS Congruence Postulate

Prove that PQW TSW.

Paragraph Proof

SOLUTION

So by the SSS Congruence Postulate, you know that

PQW TSW.

The marks on the diagram show that PQ TS,

PW TW, and QW SW.

POSTULATE

SSS AND SAS CONGRUENCE POSTULATES

POSTULATE 20 Side-Angle-Side (SAS) Congruence Postulate

Side PQ WX

Side QS XY

then PQS WXYAngle Q X

If

If two sides and the included angle of one triangle arecongruent to two sides and the included angle of a second triangle, then the two triangles are congruent.

A

S

S

1

Using the SAS Congruence Postulate

Prove that AEB DEC.

2

3 AEB DEC SAS Congruence Postulate

21

AE DE, BE CE Given

1 2 Vertical Angles Theorem

Statements Reasons

D

GA R

Proving Triangles Congruent

MODELING A REAL-LIFE SITUATION

PROVE DRA DRG

SOLUTION

ARCHITECTURE You are designing the window shown in the drawing. Youwant to make DRA congruent to DRG. You design the window so that DR AG and RA RG.

Can you conclude that DRA DRG ?

GIVEN DR AG

RA RG

ASA Postulate

Two angles and the included side of one triangle are congruent to two angles and the included side of another triangle.