Geometry (A): READ: Congruent Figures Reading

Congruent Figures

Learning Objectives

Define congruence in triangles.
Create accurate congruence statements.
Understand that if two angles of a triangle are congruent to two angles of another triangle, the remaining angles will also be congruent.
Explore properties of triangle congruence.

Introduction

Triangles are important in geometry because every other polygon can be turned into triangles by cutting them up (formally we call this adding auxiliary lines). Think of a square: If you add an auxiliary line such as a diagonal, then it is two right triangles. If we understand triangles well, then we can take what we know about triangles and apply that knowledge to all other polygons. In this chapter you will learn about congruent triangles, and in subsequent chapters you will use what you know about triangles to prove things about all kinds of shapes and figures.

Defining Congruence in Triangles

Two figures are congruent if they have exactly the same size and shape. Another way of saying this is that the two figures can be perfectly aligned when one is placed on top of the other—but you may need to rotate or flip the figures over to make them line up. When that alignment is done, the angles that are matched are called corresponding angles, and the sides that are matched are called corresponding sides.

In the diagram above, sides $\overline{AC}$ and $\overline{DE}$ have the same length, as shown by the tic marks. If two sides have the same number of tic marks, it means that they have the same length. Since $\overline{AC}$ and $\overline{DE}$ each have one tic mark, they have the same length. Once we have established that $\overline{AC} \cong \overline{DE}$ , we need to examine the other sides of the triangles. $\overline{BA}$ and $\overline{DF}$ each have two tic marks, showing that they are also congruent. Finally, as you can see, $\overline{BC} \cong \overline {EF}$ because they each have three tic marks. Each of these pairs corresponds because they are congruent to each other. Notice that the three sides of each triangle do not need to be congruent to each other, as long as they are congruent to their corresponding side on the other triangle.

When two triangles are congruent, the three pairs of corresponding angles are also congruent. Notice the tic marks in the triangles below.

We use arcs inside the angle to show congruence in angles just as tic marks show congruence in sides. From the markings in the angles we can see $\angle {A} \cong \angle {D}, \angle {B} \cong \angle {F},$ and $\angle{C} \cong \angle {E}$ .

By definition, if two triangles are congruent, then you know that all pairs of corresponding sides are congruent and all pairs of corresponding angles are congruent. This is sometimes called CPCTC: Corresponding parts of congruent triangles are congruent.

Example 1

Are the two triangles below congruent?

The question asks whether the two triangles in the diagram are congruent. To identify whether or not the triangles are congruent, each pair of corresponding sides and angles must be congruent.

Begin by examining the sides. $\overline{AC}$ and $\overline{RI}$ both have one tic mark, so they are congruent. $\overline{AB}$ and $\overline{TI}$ both have two tic marks, so they are congruent as well. $\overline{BC}$ and $\overline{RT}$ have three tic marks each, so each pair of sides is congruent.

Next you must check each angle. $\angle{I}$ and $\angle{A}$ both have one arc, so they are congruent. $\angle{T} \cong \angle{B}$ because they each have two arcs. Finally, $\angle{R}\cong\angle{C}$ because they have three arcs.

We can check that each angle in the first triangle matches with its corresponding angle in the second triangle by examining the sides. $\angle{B}$ corresponds with $\angle{T}$ because they are formed by the sides with two and three tic marks. Since all pairs of corresponding sides and angles are congruent in these two triangles, we conclude that the two triangles are congruent.

Creating Congruence Statements

We have already been using the congruence sign $\cong$ when talking about congruent sides and congruent angles.

For example, if you wanted to say that $\overline{BC}$ was congruent to $\overline{CD}$ , you could write the following statement.

$\overline{BC} \cong \overline{CD}$

In Chapter 1 you learned that the line above $BC$ with no arrows means that $BC$ is a segment (and not a line or a ray). If you were to read this statement out loud, you could say “Segment $BC$ is congruent to segment $CD$ .”

When dealing with congruence statements involving angles or triangles, you can use other symbols. Whereas the symbol $\overline{BC}$ means “segment $BC$ ,” the symbol $\angle{B}$ means “angle $B$ .” Similarly, the symbol $\triangle{ABC}$ means “triangle $ABC$ .”

When you are creating a congruence statement of two triangles, the order of the letters is very important. Corresponding parts must be written in order. That is, the angle at first letter of the first triangle corresponds with the angle at the first letter of the second triangle, the angles at the second letter correspond, and so on.

In the diagram above, if you were to name each triangle individually, they could be $\triangle{BCD}$ and $\triangle{PQR}$ . Those names seem the most appropriate because the letters are in alphabetical order. However, if you are writing a congruence statement, you could NOT say that $\triangle{BCD} \cong \triangle{PQR}$ . If you look at $\angle{B}$ , it does not correspond to $\angle{P}$ . $\angle{B}$ corresponds to $\angle{Q}$ instead (indicated by the two arcs in the angles). $\angle{C}$ corresponds to $\angle{P}$ , and $\angle{D}$ corresponds to $\angle{R}$ . Remember, you must compose the congruent statement so that the vertices are lined up for congruence. The statement below is correct.

$\triangle{BCD} \cong \triangle{QPR}$

This form may look strange at first, but this is how you must create congruence statements in any situation. Using this standard form allows your work to be easily understood by others, a crucial element of mathematics.

Example 2

Compose a congruence statement for the two triangles below.

To write an accurate congruence statement, you must be able to identify the corresponding pairs in the triangles above. Notice that $\angle{R}$ and $\angle{F}$ each have one arc mark. Similarly, $\angle{S}$ and $\angle{E}$ each have two arcs, and $\angle{T}$ and $\angle{D}$ have three arcs. Additionally, $RS = FE$ (or $\overline{RS} \cong \overline{FE}$ ), $ST = ED,$ and $RT = FD.$

So, the two triangles are congruent, and to make the most accurate statement, this should be expressed by matching corresponding vertices. You can spell the first triangle in alphabetical order and then align the second triangle to that standard.

$\triangle{RST} \cong \triangle{FED}$

Notice in example 2 that you don’t need to write the angles in alphabetical order, as long as corresponding parts match up. If you’re feeling adventurous, you could also express this statement as shown below.

$\triangle{DEF} \cong \triangle{TSR}$

Both of these congruence statements are accurate because corresponding sides and angles are aligned within the statement.

The Third Angle Theorem

Previously, you studied the triangle sum theorem, which states that the sum of the measures of the interior angles in a triangle will always be equal to $180^\circ.$ This information is useful when showing congruence. As you practiced, if you know the measures of two angles within a triangle, there is only one possible measurement of the third angle. Thus, if you can prove two corresponding angle pairs congruent, the third pair is also guaranteed to be congruent.

Third Angle Theorem

If two angles in one triangle are congruent to two angles in another triangle, then the third pair of angles are also congruent.

This may seem like an odd statement, but use the exercise below to understand it more fully.

Example 3

Identify whether or not the missing angles in the triangles below are congruent.

To identify whether or not the third angles are congruent, you must first find their measures. Start with the triangle on the left. Since you know two of the angles in the triangle, you can use the triangle sum theorem to find the missing angle. In $\triangle{WVX}$ we know

$m \angle{W} + m \angle{V} + m \angle{X} & = 180^\circ\\ 80^\circ + 35^\circ + m \angle{X} &= 180^\circ\\ 115^\circ + m \angle{X} &= 180^\circ\\ m \angle{X} &= 65^\circ\\$

The missing angle of the triangle on the left measures $65^\circ.$ Repeat this process for the triangle on the right.

$m \angle{C} + m \angle{A} + m \angle{T} & =180^\circ \\ 80^\circ + 35^\circ + m \angle{T} & = 180^\circ \\ 115^\circ + m\angle{T} &= 180^\circ \\ m\angle{T} &= 65^\circ$

So, $\angle{X} \cong \angle{T}$ . Remember that you could also identify this without using the triangle sum theorem. If two pairs of angles in two triangles are congruent, then the remaining pair of angles also must be congruent.

Lesson Summary

In this lesson, we explored congruent figures. Specifically, we have learned:

How to define congruence in triangles.
How to create accurate congruence statements.
To understand that if two angles of a triangle are congruent to two angles of another triangle, the remaining angles will also be congruent.
How to employ properties of triangle congruence.

These skills will help you understand issues of congruence involving triangles. Always look for triangles in diagrams, maps, and other mathematical representations.

Points to Consider

Now that you understand the issues inherent in triangle congruence, you will create your first congruence proof.

The following questions are for your own benefit. The answers are listed below for you to check your work and understanding.

Review Questions

Use the diagram below for problem 1.

Write a congruence statement for the two triangles above.

Exercises 2-3 use the following diagram.

Suppose the two triangles above are congruent. Write a congruence statement for these two triangles.
Explain how we know that if the two triangles are congruent, then $\angle{B} \cong \angle{Y}$ .

Use the diagram below for exercises 4-5.

Explain how we know $\angle{K} \cong \angle {W}$ .
Are these two triangles congruent? Explain why (note, “looks” are not enough of a reason!).
If you want to know the measure of all three angles in a triangle, how many angles do you need to measure with your protractor? Why?

Use the following diagram for exercises 7-10.

What is the relationship between $\angle{FGH}$ and $\angle{FGI}$ ? How do you know?
What is $m \angle{FGH}$ ? How do you know?
What property tells us $\overline{FG} \cong \overline{FG}$ ?
Write a congruence statement for these triangles.

Review Answers

$\triangle PQR \cong \triangle NML$
$\triangle BCD \cong \triangle YWX$ (Note the order of the letters is important!)
If the two triangles are congruent, then $\angle{B}$ corresponds with $\angle{Y}$ and therefore they are congruent to each other by the definition of congruence.
The third angle theorem states that if two pairs of angles are congruent in two triangles, then the third pair of angles must also be congruent
No. $\overline{KL}$ corresponds with $\overline{WX}$ but they are not the same length
You only need to measure two angles. The triangle sum theorem will help you find the measure of the third angle
$\angle{FGH}$ and $\angle{FGI}$ are supplementary since they are a linear pair
$m\angle{FGH} = 90^\circ$
The reflexive property of congruence
$\triangle{IGF} \cong \triangle{HGF}$

Last modified: Monday, June 28, 2010, 2:20 PM