Geometry/Chapter 4/Lesson 3

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Introduction[edit]

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]

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[edit]

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

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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[edit]

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

Collary 1[edit]

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

Collary 2[edit]

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

Collary 3[edit]

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

Collary 4[edit]

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

Finding x-problems[edit]