Angle Relationships Complementary, Supplementary, and Congruent Angles Supplementary angles are angles that add up to 180°. The angles below are supplementary. Complementary angles are angles that add up to 90°. The angles below are complementary. 36° + 54° = 90° Congruent angles are angles that are equal to each other. The angles below are congruent. 79° + 101° = 180° Adjacent Angles Linear Pairs “Adjacent” means “next to”. Adjacent angles are next to each other. More specifically, they share a common side between them and a common vertex. Linear pairs are a special case of adjacent angles. They are adjacent angles that form a straight line together. y These angles do not form a linear pair because the line they form together is not straight. Because linear pairs form a straight line, they add up to 180° which makes them supplementary. Note that all linear pairs are adjacent and supplementary but not all adjacent and supplementary angles are linear pairs! Vertical Angles Angles that are on opposite sides of two intersecting lines are called vertical angles. The angles must be formed by the same lines in order to be vertical. Vertical angles are always congruent (equal). To see why, consider the figure below. Angles c and a form a linear pair, so they must add up to 180°. Since angle c is 50°, angle a must be 130°. But angles a and b also form a linear pair, so if angle a is 130°, angle b must be 50°, just like angle c is! p Angles c and a form a linear pair. Angles a and b form a linear pair. Angles c and b are vertical. Writing Equations to Describe Diagrams Writing Equations \ You might encounter problems for which you will have to write equations that describe diagrams. Some key things to remember when doing this is that angles that form a right angle together add up to 90°, angles that form a straight line together add up to 180°, and angles that form a full rotation together add up to 360°. a 71° a 71° In the diagram above, angle a and the 71 degree angle form a right angle together, so we know they must add up to 90°. Here are some equations we could write… In the diagram above, angle a and the 71 degree angle form a straight line together, so we know they must add up to 180°. Here are some equations we could write… 𝑎 + 71 = 90 71 + 𝑎 = 90 𝑎 + 71 = 180 71 + 𝑎 = 180 90 − 71 = 𝑎 90 − 𝑎 = 71 180 − 71 = 𝑎 180 − 𝑎 = 71 71° a In the diagram above, angle a and the 71 degree angle form a full rotation (circle) together, so we know they must add up to 360°. Here are some equations we could write… 𝑎 + 71 = 360 71 + 𝑎 = 360 360 − 71 = 𝑎 360 − 𝑎 = 71 When an Unknown Quantity Does Not Represent an Angle Sometimes an unknown quantity written in a diagram does not represent an angle. If it is part of an expression, it must be solved for and the value plugged back into the expression to determine the angle measure. Example: What are the measures of the angles in the following diagram? 6𝑗 − 2 2𝑗 + 32 30° Solution: Begin by writing an equation that describes the diagram, then solve it. 2𝑗 + 32 + 6𝑗 − 2 + 30 = 180 8𝑗 + 60 = 180 −60 − 60 8𝑗 = 120 ÷8 ÷8 𝑗 = 15 Next, plug the value you got back into the original expressions. 2𝑗 + 32 6𝑗 − 2 2(15) + 32 30 + 32 62 6(15) − 2 90 − 2 88 The angles represented by the expressions with j are 62° and 88°.