Special Pairs of Angles Lesson 2.4 Geometry Honors Objective: Apply the definitions of complementary and supplementary angles. State and apply the theorem about vertical angles. Page 50

Lesson Focus Pairs of angles whose measures have the sum of 90 or 180 appear frequently in geometric situations. For this reason, they are given special names. This lesson studies these special angles and solves problems involving them.

Special Pairs of Angles Complementary angles (comp. s) Two angles whose measures have the sum 90. Each angle is called the complement of the other. Example: Given: 1 and 2 are complements. If m1 = 42, then m2 = 48.

Special Pairs of Angles Supplementary angles (supp. s) Two angles whose measures have the sum of 180. Each angle is called the supplement of the other. Example: Given: 1 and 2 are supplements. If m1 = 109, then m2 = 71.

EXAMPLE 1 Identify complements and supplements In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles. SOLUTION Because 32°+ 58° = 90°, BAC and RST are complementary angles. Because 122° + 58° = 180°, CAD and RST are supplementary angles. Because BAC and CAD share a common vertex and side, they are adjacent.

GUIDED PRACTICE for Example 1 In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles. 1. Because 41° + 49° = 90°, FGK and GKL are complementary angles. Because 49° + 131° = 180°, HGK and GKL are supplementary angles. Because FGK and HGK share a common vertex and side, they are adjacent.

Special Pairs of Angles Vertical Angles (Vert. s) Two angles such that the sides of one angle are opposite rays to the other sides of the other angle. When two lines intersect, they form two pairs of vertical angles.

Special Pairs of Angles Vertical Angle Theorem Vertical angles are congruent. Proof: 1 and 2 form a linear pair, so by the Definition of Supplementary Angles, they are supplementary. That is, m1 + m2 = 180°. (also, Angle Addition Postulate) 2 and 3 form a linear pair also, so m2 + m3 = 180°. Subtracting m2 from both sides of both equations, we get m1 = 180° − m2 = m3. Therefore, 1  3. You can use a similar argument to prove that 2  4.

EXAMPLE 3 Find angle measures Sports When viewed from the side, the frame of a ball-return net forms a pair of supplementary angles with the ground. Find m BCE and m ECD.

Practice Quiz Complete with always, sometimes, or never. Vertical angles _____ have a common vertex. Two right angles are _____ complementary. Right angles are _____ vertical angles. Angles A, B, and C are _____ complementary. Vertical angles _____ have a common supplement.

Practice Quiz Complete with always, sometimes, or never. Vertical angles always have a common vertex. Two right angles are never complementary. Right angles are sometimes vertical angles. Angles A, B, and C are never complementary. Vertical angles always have a common supplement.

Homework Assignment Page 53 – 54 Problems 19 – 31 odd, 32 – 35.