# Congruent Triangles

A very important topic in the study of

geometry is congruence. Thus far, we have only learned about congruent angles,

but in this section we will learn about the criteria necessary for triangles to

be congruent. Learning about congruence on this level will open the door to different

triangle congruence theorems that characterize geometry.

## Corresponding Parts

Recall that in order for lines or angles to be congruent, they had to have equal

measures. In that same way, congruent triangles are triangles with corresponding

sides and angles that are congruent, giving them the same size and shape. Because

side and angle correspondence is important, we have to be careful with the way we

name triangles. For instance, if we have ** ?ABC??DEF**, the congruence

between triangles implies the following:

*It is important to name triangles correctly to identify which segments are equal
in length and to see which angles have the same measures.*

In short, we say that two triangles are congruent if their corresponding parts (which

include lines and angles) are congruent. In a

two-column geometric proof, we could explain congruence between triangles

by saying that **“corresponding parts of congruent triangles are congruent.”**

This statement is rather long, however, so we can just write **“CPCTC”** for

short.

## Third Angles Theorem

In some instances we will need a very significant theorem to help us prove congruence

between two triangles. If we know that two angles of two separate triangles are

congruent, our inclination is to believe that their third angles are equal because

of the

Triangle Angle Sum Theorem.

This type of reasoning is correct and is a very helpful theorem to use when trying

to prove congruence between triangles. The **Third Angles Theorem** states that

if two angles of one triangle are congruent to two angles of another triangle, then

the third angles of the triangles are congruent also.

Let’s take a look at some exercises to put our knowledge of congruent triangles,

CPCTC, and the Third Angles Theorem to work.

## Examples

**(1) Which of the following expresses the correct congruence statement for the figure
below?**

**(a)**

**(b)**

**(c)**

**(d)**

**Solution:**

While it may not seem important, the order in which you list the vertices of a triangle

is very significant when trying to establish congruence between two triangles. Essentially

what we want to do is find the answer that helps us correspond the triangles’ points,

sides, and angles. The answer that corresponds these characteristics of the triangles

is **(b)**.

In answer **(b)**, we see that ** ?PQR ? ?LJK**. Let’s start off by

comparing the vertices of the triangles. In the first triangle, the point

*P*is listed first. This corresponds to the point

**on the other triangle.**

*L*We know that these points match up because congruent angles are shown at those points.

Listed next in the first triangle is point

**. We compare this to point**

*Q***of the second triangle. Again, these match up because the angles**

*J*at those points are congruent. Finally, we look at the points

**and**

*R***. The angles at those points are congruent as well.**

*K*
We can also look at the sides of the triangles to see if they correspond. For instance,

we could compare side ** PQ** to side

**. The figure indicates**

*LJ*that those sides of the triangles are congruent. We can also look at two more pairs

of sides to make sure that they correspond. Sides

**and**

*QR*

*JK*have three tick marks each, which shows that they are congruent. Finally, sides

**and**

*RP***are congruent in the figure. Thus, the correct**

*KJ*congruence statement is shown in

**(b)**.

**(2) Find the values of x and y given that ?MAS ? ?NER.**

**Solution:**

We have two variables we need to solve for. It would be easiest to use the *16x*

to solve for ** x** first (because it is a single-variable expression),

as opposed to using the side

**, would require us to try to solve**

*NR*for

**and**

*x***at the same time. We must look for the angle**

*y*that correspond to

**so we can set the measures equal to each other.**

*?E*The angle that corresponds to

**is**

*?E***, so we get**

*?A*

Now that we have solved for ** x**, we must use it to help us solve for

**. The side that**

*y***corresponds to is**

*RN***,**

*SM*so we go through a similar process like we did before.

Now we substitute ** 7** for

**to solve for**

*x***:**

*y*

We have finished solving for the desired variables.

**(3) Given:**

**
**

**Prove:**

**Solution:**

To begin this problem, we must be conscious of the information that has been given

to us. We know that two pairs of sides are congruent and that one set of angles

is congruent. In order to prove the congruence of ** ?RQT** and

**,**

*?SQT*we must show that the three pairs of sides and the three pairs of angles are congruent.

Since ** QS** is shared by both triangles, we can use the

**Reflexive Property**

to show that the segment is congruent to itself. We have now proven congruence between

the three pairs of sides. The congruence of the other two pairs of sides were already

given to us, so we are done proving congruence between the sides.

Now we must show that all angles are congruent within the triangles. One pair has

already been given to us, so we must show that the other two pairs are congruent.

It has been given to us that ** QT** bisects

**. By the**

*?RQS*definition of an angle bisector, we know that two equivalent angles exist at vertex

**. The final pairs of angles are congruent by the**

*Q***Third Angles Theorem**

(since the other two pairs of corresponding angles of the triangles were congruent).

We conclude that the triangles are congruent because corresponding parts of congruent

triangles are congruent. The two-column geometric proof that shows our reasoning

is below.

**(4) Given:**

**Prove:**

**Solution:**

We are given that the three pairs of corresponding sides are congruent, so we do

not have to worry about this part of the problem; we only need to worry about proving

congruence between corresponding angles.

We are only given that one pair of corresponding angles is congruent, so we must

determine a way to prove that the other two pairs of corresponding angles are congruent.

We do this by showing that ** ?ACB** and

**are vertical**

*?ECD*angles. So, by the

**Vertical Angles Theorem**, we know that they are congruent

to each other. Now that we know that two of the three pairs of corresponding angles

of the triangles are congruent, we can use the

**Third Angles Theorem**. This

theorem states that if we have two pairs of corresponding angles that are congruent,

then the third pair must also be congruent.

Since all three pairs of sides and angles have been proven to be congruent, we know

the two triangles are congruent by **CPCTC**. The two-column geometric proof

that shows our reasoning is below.