Learning Resources

Lesson

If two triangles are congruent, what can we conclude?

We want to be able to complete the statement: "If two triangles are congruent then ????"

That is the statement we will focus on in the first part of this lesson. By definition, any two figures are congruent if they are identical in shape and size. For triangles this means the three corresponding/matching sides are congruent AND the three corresponding/matching angles are congruent.

Consider the two triangles below.:  

If D ABC @ D PQR then

Sides are congruent AB @ PQ BC @ QR AC @ PR

Angles are congruent ÐABC  @ ÐPQR ÐBCA  @ ÐQRP ÐCAB  @ ÐRPQ

Note particularly the order in which the triangle congruence is written, as this tells which sides and angles match. For example, without having a diagram drawn we know that:

 D DEF @  D XYZ   Þ  DE @ XY   (matching sides)

 D DEF @  D XYZ   Þ  EF @ YZ   (matching sides)

 D DEF @  D XYZ   Þ  DF @ XZ   (matching sides)

AND

D DEF @  D XYZ   Þ  ÐD @ ÐX   (matching angles)

D DEF @  D XYZ   Þ  ÐE @ ÐY   (matching angles)

D DEF @  D XYZ   Þ  ÐF @ ÐZ   (matching angles)

 

Note

The mathematical symbol   Þ means "implies", so writing
D DEF @  D XYZ   Þ  DE @ XY is the same as saying if D DEF @  D XYZ, then DE @ XY.

What information is necessary to conclude that two triangles are congruent?

We want to be able to complete the statement: " If in two triangles ????, then two triangles are congruent."

From the definition of congruence on the previous page and also in the margin on page 212 of your text book, we can say the following:

If in two triangles the three corresponding sides are congruent AND the three corresponding angles are congruent, then the triangles are congruent.

Thus if six pieces of information are known (the three sides AND the three angles) we know by definition that the triangles are congruent. However, it is not necessary to have all six pieces of information. In fact, to conclude that triangles are congruent, we only need 3 pieces of information, not 6, provided they are the correct three pieces. 

What three pieces do we need?

First consider three pieces of information that would not be sufficient to say two triangles are congruent.  Knowing only the three angles would not be sufficient as is shown in the diagram below:

                

ÐA  @ ÐP,  ÐB  @ ÐQ  and   ÐC  @ ÐR  but DABC is NOT congruent to DPQR as they are obviously not the same size.

There are five different possibilities for the three pieces that are sufficient and these are shown quite nicely in your text on pages 212 & 213. Only one of these five is discussed in detail here. For the other possibilities, all of which are called postulates, go to your page 213 in your text.

Note: A postulate is something that is assumed to be true, usually because it is intuitively obvious. It is then used as a basis for argument in a logical reasoning process or proof.

The Side-Side-Side Congruency Postulate (SSS)

This postulate states that if the three sides of one triangle are congruent to the matching three sides of another triangle, then the triangles are congruent. We have only three pieces of information, viz. the three sides and we are claiming that is sufficient to say the triangles are congruent. In other words, if the three sides are congruent, then the three angles must also be congruent.

Let's construct a triangle whose three sides are congruent to the corresponding three sides of a given triangle and see if in fact the angles turn out to be congruent.

The other postulates are referred to as the SAS, ASA, SAA, and HL postulates. They are discussed on page 213 in your text. Don't be too concerned about a justification similar to what we have just done for these postulates. It is more important to know what they say and be able to apply them to other proofs. It is the application of these postulates to which the rest of this lesson is devoted.

In your text book on pages 214 and 215 are several examples of how these postulates can be used to prove new properties and facts about triangles. Read them carefully before beginning the assigned activities.

Two other examples are done in this lesson to serve as a guide for the assigned work. Simply click on each example in turn to see the presentation.

• Example 1
• Example 2

In this lesson you were introduced to and should now have a thorough understanding of the following:

1. SSS Congruency Postulate:
   If the three sides of one triangle are congruent to the three sides of a second triangle, then the triangles are congruent.
2. SAS Congruency postulate:
   If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent.
3. ASA Congruency Postulate:
   If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent.
4. SAA Congruency Postulate:
   If two angles and a non-included side of one triangle are congruent to two angles and a non-included side of a second triangle, then the triangles are congruent.
5. HL Congruency Postulate:
   If the hypotenuse and a leg of one right triangle is congruent to the hypotenuse and a leg of a second right triangle, then the triangles are congruent.
6. Proof:
   Be able to apply all the congruency postulates to prove or justify other properties about triangles.

Activity

1. Read through the examples of proofs on page 214 - 216 in your text.
2. Complete the Focus Questions 22 - 24 on pages 216 & 217.
3. Do the CYU Questions 25 & 26 on page 217.

When you have completed these questions, ask your on-site teacher to get the solutions for you from the Teacher's Resource Binder and check them against your answers. After you do this, if there is something you had trouble with and still do not understand, contact your on-line teacher for help.

Test Yourself

Given:     FC || ED                FC = ED                AB = CD   Prove:     FA || EB

Solution