Geometry/Chapter 4/Lesson 3

Introduction

This lesson will teach you the following: Exterior Angles, collaries and you will do more finding "x" in a triangle problems. Let's begin!

Exterior Angles

An exterior angle and its sum to help better understand Theorem 2-2

An exterior angle of a polygon is formed by one side of a triangle and an extension of another side. The two non-adjacent angles, that every exterior angle has, are called its remote interior angles.

Theorem 2-2

Theorem 2-2 m∠1 = m∠2 + m∠3 The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles

thumbless

Look at the picture to our left... We have several "x"s to solve. Here, we will, step-by-step, go through the problems and find the sums of numbers 1, 2, 3, 4 and 5.

1

"1" is an exterior angle, and we are given the two remote interior angles, which are 50° and 78°. If we add up 50+78 (since according to Theorem 2-2, the measure of each exterior angle's sum = its two remote interior angles' sum), we will get 128°. So, by using the Theorem 2-2, we have now gotten our answer: 128°.
1 = 128°

2

128° and "2" are a linear pair, so we simply do the following problem: 180 - 128... which is 52. 2 = 52°.

3

Here, we have the Exterior Angle Theorem (Theorem 2-2) backward... now, instead of finding the sum of the exterior angle, we are trying to find one of the exterior angle's remote interior angles! Now that we have:

• Our exterior angle = 120°
• One of our remote interior angles = 52

So we simply do 120 - 52, which is 68. 3 = 68°.

4

"4" and 120° form a linear pair, so we have to do: 180 - 120, which equals 60. 4 = 60°.

5

Exterior Angle Theorem: 60 + 56 = 116. 5 = 116.

Collaries

A collary is a statement can be proved easily by applying a theorem.

Collary 1

thumbless

• If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.

Collary 2

• Each angle of an equiangular triangle has a measure of 60 degrees.

Collary 3

• In a triangle, there can be, at most, 1 right angle or obtuse angle.

Collary 4

• The acute angles of a right triangle are complementary (= 90).

Finding x-problems

← Lesson 2

Geometry/Chapter 4

Lesson 4 →