4.3 Proving Triangles Congruent: SSS and SAS

GOAL: PROVE THAT TRIANGLES ARE CONGRUENT USING THE SSS AND SAS CONGRUENCE POSTULATES.

Activities:1. Open SSS. Complete steps 1 and 2 and answer Q1

and Q2 on the Triangle Congruence activity sheet. 2. Open SAS. Complete step 3 and answer Q3 on the

Triangle Congruence activity sheet.3. View Lesson 4.3. Work out all examples and take

notes.4. Summary

a. In your notes, explain the two concepts explored in class today relating measures of sides and angles in triangles.

b. Would SSA (two sides and a non-included angle of one triangle congruent to two sides and a non-included angle of another) work? Explain.

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 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 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 are congruent

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

2

3

4

5

6 SAS Congruence Postulate DRA DRG

1

Proving Triangles Congruent

GivenDR AG

If 2 lines are , then they form 4 right angles.

DRA and DRGare right angles.

All right angles are congruent DRA DRG

GivenRA RG

Reflexive Property of CongruenceDR DR

Statements Reasons

D

GA R

GIVEN

PROVE DRA DRG

DR AG

RA RG

Congruent Triangles in a Coordinate Plane

AC FH

AB FGAB = 5 and FG = 5

SOLUTION

Use the SSS Congruence Postulate to show that ABC FGH.

AC = 3 and FH = 3

Congruent Triangles in a Coordinate Plane

d = (x 2 – x1 ) 2 + ( y2 – y1 )

2

= 3 2 + 5

2

= 34

BC = (– 4 – (– 7)) 2 + (5 – 0 )

2

d = (x 2 – x1 ) 2 + ( y2 – y1 )

2

= 5 2 + 3

2

= 34

GH = (6 – 1) 2 + (5 – 2 )

2

Use the distance formula to find lengths BC and GH.

Congruent Triangles in a Coordinate Plane

BC GH

All three pairs of corresponding sides are congruent, ABC FGH by the SSS Congruence Postulate.

BC = 34 and GH = 34