Geometry (A): READ: Isosceles and Equilateral Triangles Reading

Isosceles and Equilateral Triangles

Learning Objectives

Prove and use the Base Angles Theorem.
Prove that an equilateral triangle must also be equiangular.
Use the converse of the Base Angles Theorem.
Prove that an equiangular triangle must also be equilateral.

Introduction

As you can imagine, there is more to triangles than proving them congruent. There are many different ways to analyze the angles and sides within a triangle to understand it better. This chapter addresses some of the ways you can find information about two special triangles.

Base Angles Theorem

An isosceles triangle is defined as a triangle that has at least two congruent sides. In this lesson you will prove that an isosceles triangle also has two congruent angles opposite the two congruent sides. The congruent sides of the isosceles triangle are called the legs of the triangle. The other side is called the base and the angles between the base and the congruent sides are called base angles. The angle made by the two legs of the isosceles triangle is called the vertex angle.

The Base Angles Theorem states that if two sides of a triangle are congruent, then their opposite angles are also congruent. In other words, the base angles of an isosceles triangle are congruent. Note, this theorem does not tell us about the vertex angle.

Example 1

Which two angles must be congruent in the diagram below?

The triangle in the diagram is an isosceles triangle. To find the congruent angles, you need to find the angles that are opposite the congruent sides.

This diagram shows the congruent angles. The congruent angles in the triangle are $\angle{XYW}$ and $\angle{XWY}$ .

So, how do we prove the base angles theorem? Using congruent triangles.

Given: Isosceles $\triangle{ABC}$ with $\overline{AB} \cong \overline{AC}$

Prove $\angle{B} \cong \angle{C}$

Statement	Reason
1. $\triangle{ABC}$ is isosceles with $\overline{AB} \cong \overline{AC}$ .	1. Given
2. Construct Angle Bisector $\overline{AD}$	2. Angle Bisector Postulate
3. $\angle{BAD} \cong \angle{CAD}$	3. Definition of Angle Bisector
4. $\overline{AD} \cong \overline{AD}$	4. Reflexive Property

5. $\triangle{ABD} \cong \triangle{ACD}$	5. SAS Postulate
6. $\angle{B} \cong \angle{C}$	6. Definition of congruent triangles (all pairs of corresponding angles are congruent)

Equilateral Triangles

The base angles theorem also applies to equilateral triangles. By definition, all sides in an equilateral triangle have exactly the same length.

Because of the base angles theorem, we know that angles opposite congruent sides in an isosceles triangle are congruent. So, if all three sides of the triangle are congruent, then all of the angles are congruent as well.

A triangle that has all angles congruent is called an equiangular triangle. So, as a result of the base angles theorem, you can identify that all equilateral triangles are also equiangular triangles.

Converse of the Base Angles Theorem

As you know, some theorems have a converse that is also true. Recall that a converse identifies the “backwards,” or reverse statement of a theorem. For example, if I say, “If I turn a faucet on, then water comes out,” I have made a statement. The converse of that statement is, “If water comes out of a faucet, then I have turned the faucet on.” In this case the converse is not true. For example the faucet may have a drip. So, as you can see, converse statements are sometimes true, but not always.

The converse of the base angles theorem is always true. The base angles theorem states that if two sides of a triangle are congruent the angles opposite them are also congruent. The converse of this statement is that if two angles in a triangle are congruent, then the sides opposite them will also be congruent. You can use this information to identify isosceles triangles in many different circumstances.

Example 2

Which two sides must be congruent in the diagram below?

$\triangle{WXY}$ has two congruent angles. By the converse of the base angles theorem, it is an isosceles triangle. To find the congruent sides, you need to find the sides that are opposite the congruent angles.

This diagram shows arrows pointing to the congruent sides. The congruent sides in this triangle are $\overline{XY}$ and $\overline{XW}$ .

The proof of the converse of the base angles theorem will depend on a few more properties of isosceles triangles that we will prove later, so for now we will omit that proof.

Equiangular Triangles

Earlier in this lesson, you extrapolated that all equilateral triangles were also equiangular triangles and proved it using the base angles theorem. Now that you understand that the converse of the base angles theorem is also true, the converse of the equilateral/equiangular relationship will also be true.

If a triangle has three congruent angles, it is be equiangular. Since congruent angles have congruent sides opposite them, all sides in an equiangular triangle will also be congruent. Therefore, every equiangular triangle is also equilateral.

Lesson Summary

In this lesson, we explored isosceles, equilateral, and equiangular triangles. Specifically, we have learned to:

Prove and use the Base Angles Theorem.
Prove that an equilateral triangle must also be equiangular.
Use the converse of the Base Angles Theorem.
Prove that an equiangular triangle must also be equilateral.

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

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

Review Questions

Sketch and label an isosceles $\triangle ABC$ with legs $\overline{AB}$ and $\overline{BC}$ that has a vertex angle measuring $118^\circ.$
What is the measure of each base angle in $\triangle ABC$ from 1?
Find the measure of each angle in the triangle below:
below is equilateral. If bisects , find:
1. $m\angle{EUL}$
2. $m\angle{UEL}$
3. $m\angle{ELQ}$
Which of the following statements must be true about the base angles of an isosceles triangle ?
1. The base angles are congruent.
2. The base angles are complementary.
3. The base angles are acute.
4. The base angles can be right angles.
One of the statements in 5 is possible (i.e., sometimes true), but not necessarily always true. Which one is it? For the statement that is always false draw a sketch to show why.

7-13: In the diagram below, $m_1 \| m_2$ . Use the given angle measure and the geometric markings to find each of the following angles.

$a =$ _____
$b =$ _____
$c =$ _____
$d =$ _____
$e =$ _____
$f =$ _____
$g =$ _____

Review Answers

Each base angle in $\triangle ABC$ measures $31^\circ$
$m\angle{R} = 64^\circ$ and $m\angle{Q} = 52^\circ$
1. $m\angle{EUL} = 90^\circ$ ,
2. $m\angle{UEL} = 30^\circ,$
3. $m\angle{ELQ} = 60^\circ$
a. and c. only.
b. is possible if the base angles are $45^\circ.$ When this happens, the vertex angle is $90^\circ.$ d. is impossible because if the base angles are right angles, then the “sides” will be parallel and you won’t have a triangle.
$a = 46^\circ$
$b = 88^\circ$
$c = 46^\circ$
$d = 134^\circ$
$e = 46^\circ$
$f = 67^\circ$
$g = 67^\circ$

Last modified: Monday, June 28, 2010, 3:44 PM