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 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
 The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles

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" 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° 128° and "2" are a linear pair, so we simply do the following problem: 180 - 128... which is 52. 2 = 52°. 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: So we simply do 120 - 52, which is 68. 3 = 68°. "4" and 120° form a linear pair, so we have to do: 180 - 120, which equals 60. 4 = 60°. Exterior Angle Theorem: 60 + 56 = 116. 5 = 116.
 * 1
 * 2
 * 3
 * Our exterior angle = 120°
 * One of our remote interior angles = 52
 * 4
 * 5

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

Collary 1

 * 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).