Geometry/Chapter 4/Lesson 3
Introduction[edit  edit source]
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[edit  edit source]
An exterior angle of a polygon is formed by one side of a triangle and an extension of another side. The two nonadjacent angles, that every exterior angle has, are called its remote interior angles.
Theorem 22[edit  edit source]
Theorem 22

Look at the picture to our left... We have several "x"s to solve. Here, we will, stepbystep, 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 22, the measure of each exterior angle's sum = its two remote interior angles' sum), we will get 128°. So, by using the Theorem 22, 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 22) 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[edit  edit source]
A collary is a statement can be proved easily by applying a theorem.
Collary 1[edit  edit source]
 If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.
Collary 2[edit  edit source]
 Each angle of an equiangular triangle has a measure of 60 degrees.
Collary 3[edit  edit source]
 In a triangle, there can be, at most, 1 right angle or obtuse angle.
Collary 4[edit  edit source]
 The acute angles of a right triangle are complementary (= 90).