Geometry (A): READ: Using Congruent Triangles Reading

Using Congruent Triangles

Learning Objectives

Apply various triangles congruence postulates and theorems.
Know the ways in which you can prove parts of a triangle congruent.
Find distances using congruent triangles.
Use construction techniques to create congruent triangles.

Introduction

As you can see, there are many different ways to prove that two triangles are congruent. It is important to know all of the different way that can prove congruence, and it is important to know which combinations of sides and angles do not prove congruence. When you prove properties of polygons in later chapters you will frequently use

Congruence Theorem Review

As you have studied in the previous lessons, there are five theorems and postulates that provide different ways in which you can prove two triangles congruent without checking all of the angles and all of the sides. It is important to know these five rules well so that you can use them in practical applications.

Name	Corresponding congruent parts	Does it prove congruence?
SSS	Three sides	Yes
SAS	Two sides and the angle between them	Yes
ASA	Two angles and the side between them	Yes
AAS	Two angles and a side not between them	Yes
HL	A hypotenuse and a leg in a right triangle	Yes
AAA	Three angles	No—it will create a similar triangle, but not of the same size
SSA	Two sides and an angle not between them	No—this can create more than one distinct triangle

When in doubt, think about the models we created. If you can construct only one possible triangle given the constraints, then you can prove congruence. If you can create more than one triangle within the given information, you cannot prove congruence.

Example 1

What rule can prove that the triangles below are congruent?

A. SSS

B. SSA

C. ASA

D. AAS

The two triangles in the picture have two pairs of congruent angles and one pair of corresponding congruent sides. So, the triangle congruence postulate you choose must have two $A's$ (for the angles) and one $S$ (for the side). You can eliminate choices $A$ and $B$ for this reason. Now that you are deciding between choices $C$ and $D$ , you need to identify where the side is located in relation to the given angles. It is adjacent to one angle, but it is not in between them. Therefore, you can prove congruence using AAS. The correct answer is D.

Proving Parts Congruent

It is one thing to identify congruence when all of the important identifying information is provided, but sometimes you will have to identify congruent parts on your own. You have already practiced this in a few different ways. When you were testing SSS congruence, you used the distance formula to find the lengths of sides on a coordinate grid. As a review, the distance formula is shown below.

$\text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

You can use the distance formula whenever you are examining shapes on a coordinate grid.

When you were creating two-column and flow proofs, you also used the reflexive property of congruence. This property states that any segment or angle is congruent to itself. While this may sound obvious, it can be very helpful in proofs, as you saw in those examples.

You may be tempted to use your ruler and protractor to check whether two triangles are congruent. However, this method does not necessarily work because not all pictures are drawn to scale.

Example 2

How could you prove $\triangle{ABC} \cong \triangle{DEC}$ in the diagram below?

We can already see that $\overline{BC} \cong \overline{CE}$ and $\overline{AC} \cong \overline{CD}$ . We may be able to use SSS or SAS to show the triangles are congruent. However, to use SSS, we would need $\overline{AB} \cong \overline{DE}$ and there is no obvious way to prove this. Can we show that two of the angles are congruent? Notice that $\angle{BCA}$ and $\angle{ECD}$ are vertical angles (nonadjacent angles made by the intersection of two lines—i.e., angles on the opposite sides of the intersection).

The Vertical Angle Theorem states that all vertical angles are also congruent. So, this tells us that $\angle{BCA} \cong \angle{ECD}$ . Finally, by putting all the information together, you can confirm that $\triangle{ABC} \cong \triangle{DEC}$ by the SAS Postulate.

Finding Distances

One way to use congruent triangles is to help you find distances in real life—usually using a map or a diagram as a model.

When using congruent triangles to identify distances, be sure you always match up corresponding sides. The most common error on this type of problem involves matching two sides that are not corresponding.

Example 3

The map below shows five different towns. The town of Meridian was given its name because it lies exactly halfway between two pairs of cities: Camden and Grenata, and Lowell and Morsetown.

Using the information in the map, what is the distance between Camden and Lowell?

The first step in this problem is to identify whether or not the marked triangles are congruent. Since you know that the distance from Camden to Meridian is the same as Meridian to Grenata, those two sides are congruent. Similarly, since the distance from Lowell to Meridian is the same as Meridian to Morsetown, those two sides are also a congruent pair. The angles between these lines are also congruent because they are vertical angles.

So, by the SAS postulate, these two triangles are congruent. This allows us to find the distance between Camden and Lowell by identifying its corresponding side on the other triangle. Because they are both opposite the vertical angle, the side connecting Camden and Lowell corresponds to the side connecting Morsetown and Grenata. Since the triangles are congruent, these corresponding sides will also be congruent to each other. Therefore, the distance between Camden and Lowell is five miles.

This use of the definition of congruent triangles is one of the most powerful tools you will use in geometry class. It is often abbreviated as CPCTC, meaning Corresponding Parts of Congruent Triangles are Congruent.

Constructions

Another important part of geometry is creating geometric figures through construction. A construction is a drawing that is made using only a straightedge and a compass—you can think of construction as a special game in geometry in which we make figures using only these tools. You may be surprised how many shapes can be made this way.

Example 4

Use a compass and straightedge to construct the perpendicular bisector of the segment below.

Begin by using your compass to create an arc with the same distance from a point as the segment.

Repeat this process on the opposite side.

Now draw a line through the two points of intersections. This forms the perpendicular bisector.

Draw segments connecting the points on the bisector to the original endpoints.

Knowing that the center point is the midpoint of both line segments and that all angles formed around point $M$ are right angles, you can prove that all four triangles created are congruent by the SAS rule.

Lesson Summary

In this lesson, we explored applications triangle congruence. Specifically, we have learned to:

Identify various triangles congruence postulates and theorems.
Use the fact that corresponding parts of congruent triangles are congruent.
Find distances using congruent triangles.
Use construction techniques to create congruent triangles.

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

You now know all the different ways in which you can prove two triangles congruent. In the next chapter you’ll learn more about isosceles and equilateral triangles.

Last modified: Monday, June 28, 2010, 3:35 PM