geometry a bellwork

Holt Geometry

4-4 Triangle Congruence: SSS and SAS

Geometry A Bellwork3) Write a congruence statement that indicates that the two triangles are congruent.

A

B

C

D

Holt Geometry

4-4 Triangle Congruence: SSS and SAS4-4 Triangle Congruence: SSS and SAS

Holt Geometry

Holt Geometry

4-4 Triangle Congruence: SSS and SAS

In Lesson 4-1, you proved triangles congruent by showing that all six pairs of corresponding parts were congruent.

Now we will learn of several shortcuts to prove triangles congruent. We can prove them congruent using SIDE-SIDE-SIDE or SIDE-ANGLE-SIDE.

Holt Geometry

4-4 Triangle Congruence: SSS and SAS

4-1

Holt Geometry

4-4 Triangle Congruence: SSS and SAS

Adjacent triangles share a side, so you can apply the Reflexive Property to get a pair of congruent parts.

Remember!

Holt Geometry

4-4 Triangle Congruence: SSS and SAS

Example 1: Using SSS to Prove Triangle Congruence

Use SSS to explain why ∆ABC ∆DBC.

It is given that AC DC and that AB DB. By the Reflexive Property of Congruence, BC BC. Therefore ∆ABC ∆DBC by SSS.

Holt Geometry

4-4 Triangle Congruence: SSS and SAS

Check It Out! Example 1

Use SSS to explain why ∆ABC ∆CDA.

It is given that AB CD and BC DA.

By the Reflexive Property of Congruence, AC CA.

So ∆ABC ∆CDA by SSS.

Holt Geometry

4-4 Triangle Congruence: SSS and SAS

An included angle is an angle formed by two adjacent sides of a polygon.

B is the included angle between sides AB and BC.

Holt Geometry

4-4 Triangle Congruence: SSS and SAS

4-2

Holt Geometry

4-4 Triangle Congruence: SSS and SAS

The letters SAS are written in that order because the congruent angles must be between pairs of congruent corresponding sides.

Caution

Holt Geometry

4-4 Triangle Congruence: SSS and SAS

Check It Out! Example 2

Use SAS to explain why ∆ABC ∆DBC.

It is given that BA BD and ABC DBC. By the Reflexive Property of , BC BC. So ∆ABC ∆DBC by SAS.

Holt Geometry

4-4 Triangle Congruence: SSS and SAS

Example 3A: Verifying Triangle Congruence

Determine if we know these triangles are congruent by SSS or SAS.

∆MNO ∆PQR.

∆MNO ∆PQR by SSS.

PQ MN, QR NO, PR MO

Holt Geometry

4-4 Triangle Congruence: SSS and SAS

Example 3B: Verifying Triangle Congruence

∆STU ∆VWX.

∆STU ∆VWX by SAS.

ST VW, TU WX, and T W.

Determine if we know these triangles are congruent by SSS or SAS.

Holt Geometry

4-4 Triangle Congruence: SSS and SAS

Check It Out! Example 3

∆ADB ∆CDB by SAS.

Determine if we know these triangles are congruent by SSS or SAS.

Holt Geometry

4-4 Triangle Congruence: SSS and SAS

Example 4: Proving Triangles Congruent

∆ABD ∆CDB

Determine if we know these triangles are congruent by SSS or SAS.

∆ABD ∆CDB by SAS.

Holt Geometry

4-4 Triangle Congruence: SSS and SAS

Lesson Quiz: Part I

Can SSS or SAS or none be used to prove the triangles congruent?

1.

2.

none

SSS

Holt Geometry

4-4 Triangle Congruence: SSS and SAS

Example 2: Engineering Application

The diagram shows part of the support structure for a tower. Use SAS to explain why ∆XYZ ∆VWZ.

It is given that XZ VZ and that YZ WZ. By the Vertical s Theorem. XZY VZW. Therefore ∆XYZ ∆VWZ by SAS.