Name: ______________________ Date: __________________

Foundations of Mathematics 11

Chapter 2- Angles & Triangles 2.5 Exploring Congruent Triangles

Goal: Learn to determine the minimum amount of information needed to prove that two

triangles are congruent.

Conditions for Congruency in Triangles

There are minimum sets of angle and side measurements that, if known, allow you to

conclude that two triangles are congruent.

If three pairs of corresponding sides are equal, then the triangles are congruent. This is

known as side-side-side congruence, or SSS.

List the corresponding equal sides:

If two pairs of corresponding sides and the contained angles are equal, then the

triangles are congruent. This is known as the side-angle-side congruence or SAS

If two pairs of corresponding angles and the contained sides are equal, then the

triangles are congruent. This is known as the angle-side-angle congruence or ASA

List the corresponding equal sides:

A

Let's try a few of these.......

Given: AD BE, D E, DC EC

Prove: ADC BEC

Given: AB CB, AD CD

Prove: ABD CBD

Statement Reason

1. (S) AD BE 1.

2. (A) D E 2.

3. (S) DC EC 3.

4. ADC BEC 4.

Statement Reason

1. (S) AB CB 1.

2. (S) AD CD 2.

3. (S) BD BD 3.

4. ABD CBD 4.

The process of proof for these types of problems is to basically list the 3 key items that show

congruence. So if you can identify the congruence postulate (SSS, SAS, ASA) before you begin it is

usually quite easy to complete the proof.

GIVEN: DAC BCA, DA BC

PROVE: DAC BCA

Statements Reasons

1. (A) DAC BCA 1.

2. (S) DA BC 2.

3. (S) AC AC 3.

4. DAC BAC 4.

GIVEN: ACD CAB , AD || CB

PROVE: DAC BCA

Statements Reasons

1. (A) ACD CAB,

AD || CB 1.

2. (S) AC CA 2.

3. (A) CAB ACB 3.

4. DAC BCA 4.

Hopefully you can see that these proofs are quite straight forward and usually only require you to fill

in a line or two beyond the given.

In all the previous proofs you were given direct statements about the congruence of angles or of sides.

In the below proof you will be given information about a midpoint that will need to be translated into

information about sides.

GIVEN: C is the midpoint of BD, 1 2

PROVE: BCA DCA

Statements Reasons

1. (A) 1 2 1.

2. C is the midpoint of BD 2.

3. (S) BC DC 3.

4. (S) AC AC 4.

5. BCA DCA 5.

In all proofs so far in this chapter we have been working towards congruence of two triangles but

sometimes we want to show or prove that a side or an angle is congruent. To do this we FIRST prove

the triangles to be congruent and then of course by the definition of congruent triangles (CPCTC -

Corresponding Parts of Congruent Triangles are Congruent (remember I told you we would use

this idea)) the corresponding parts are congruent. Let me show you how this works.......

GIVEN: ABE DBE, AEB DEB

PROVE: AB DB

Statements Reasons

1. (A) ABE DBE 1.

2. (A) AEB DEB 2.

3. (S) BE BE 3.

4. CAD BAD 4.

5. AB DB 5.

Summary

I hope that you don't feel like these proofs are too difficult... Really they follow a simple logical

formula, list the given, state any other sides or angles that you know are congruent and then determine

if it is SSS, SAS or ASA.

You also saw what we do to 'translate' givens that are not directly side or angle congruence as well as

what to do if the 'prove' line has to do with sides and angles.

Assignment: Page 106 #1 - 4