Lesson #9 Geometry: Angle Facts

With just a few basic facts, we can determine many angles in a geometric figure. Angles are typically measured in degrees, with 360° representing a full circle. If you are given any angles in a figure, you should first attempt to identify the measure of other angles by using the following four facts. One word of caution: never assume that geometric figures are drawn accurately. Be careful to use only the given information to solve a geometry problem.

 Fact 1. A small square drawn between two lines indicates a right angle (that is, the two lines are perpendicular. That is, they cross at a 90° angle, exactly one-fourth of a full circle. A 90° angle is also called a right angle. Fact 2. Any line crossing two parallel lines forms equal angles. Each angle marked x in figure 2 has the same measure. Fact 3. Adjacent angles where two lines cross add up to 180°. Such pairs of angles are called supplementary angles. Fact 4. For any triangle, the three angles total 180°.

Example 1. What is the measure of the unknown angle in triangle ABC?

 Call the missing angle x. By Fact 4, the three angles must add up to 180°. So we have the equation x + 80° + 35° = 180°. Solve for x to get x = 65°.

Example 2. What angle is supplementary to 27°?

Let x represent the supplementary angle. Since supplementary angles add up to 180° (Fact 3), we must have x + 27° = 180°. Solve for x to get x = 143°.

Example 3. Find all of the other angles in the diagram below.

 Since angle a is supplementary to 38°, we have a + 38° = 180°. Thus a = 142°. Then, by Fact 2, we find b = 38° and c = 142° (opposite angles are equal).

Example 4. What is the measure of the angle x below?

 The angle a is supplementary to 160°, so a = 20°. Then a + b + 90° = 180° (within the same triangle). Substitute a = 20° and solve for b to get b = 70°. Finally, angles x and b are supplementary, so x + 70° = 180°, which gives x = 110°.

Exercises

 1) In the diagram at right, which angle is the same size as z? (There is exactly one)

In each of the following diagrams, find the measure of the supplementary angle x.

 2) 3) 4)

In each triangle below, find the measure of the third angle.

 5) 6) 7) 8) 9) 10)

11) A triangle is called equilateral if it has three sides all of equal length. Its angles must also be equal in size. What is the measure of each angle in an equilateral triangle?

In each of the figures below, find the measure of the angle x.

 12) 13) 14)

A triangle is called isosceles if it has two sides of equal length. A geometric consequence is that it also has two equal angles, which are called the base angles. Find the measure of the base angles in the following isosceles triangles. Hint: Let x represent each base angle. The base angles are always opposite the two equal sides.

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 A right triangle is one that contains a 90° angle. By Fact 4, the three angles must add up to 180°. Since one of these is 90° , the other two angles must add up to 90° . If the two angles are labeled x and y, then we have the equation: x + y = 90°.

Find the unknown angle in each of the right triangles below.

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Check the answers to Lesson 9 exercises

Go on to Lesson 10