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Geometry A Bellwork

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Geometry A Bellwork

A

3) Write a congruence statement that indicates that the two triangles are congruent.

D

B

C

4-4

Triangle Congruence: SSS and SAS

Holt Geometry

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.

4-1

Remember!

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

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

Example 1: Using SSS to Prove Triangle Congruence

Use SSS to explain why ∆ABC ∆DBC.

It is given that AB CD and BC DA.

By the Reflexive Property of Congruence, AC CA.

So ∆ABC ∆CDA by SSS.

Check It Out! Example 1

Use SSS to explain why

∆ABC ∆CDA.

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

B is the included angle between sides AB and BC.

4-2

Caution

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

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

Check It Out! Example 2

Use SAS to explain why ∆ABC ∆DBC.

PQ MN, QR NO, PR MO

Example 3A: Verifying Triangle Congruence

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

∆MNO ∆PQR.

∆MNO ∆PQR by SSS.

ST VW, TU WX, and T W.

Example 3B: Verifying Triangle Congruence

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

∆STU ∆VWX.

∆STU ∆VWX by SAS.

Check It Out! Example 3

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

∆ADB ∆CDB by SAS.

Example 4: Proving Triangles Congruent

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

∆ABD ∆CDB

∆ABD ∆CDB by SAS.

Lesson Quiz: Part I

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

1.

none

2.

SSS

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

Example 2: Engineering Application

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