Special Pairs of AnglesSpecial Pairs of AnglesLesson 2.4Lesson 2.4

Geometry HonorsGeometry Honors

Objective: Apply the definitions of Objective: Apply the definitions of complementary and supplementary angles.complementary and supplementary angles.State and apply the theorem about vertical State and apply the theorem about vertical

angles.angles.Page 50Page 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 m1 = 42, then m2 = 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 m1 = 109, then m2 = 71.

EXAMPLE 1 Identify complements and supplements

SOLUTION

In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles.

Because 122° + 58° = 180°, CAD and RST are supplementary angles.

Because BAC and CAD share a common vertex and side, they are adjacent.

Because 32°+ 58° = 90°, BAC and RST are complementary angles.

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 FGK and HGK share a common vertex and side, they are adjacent.

Because 49° + 131° = 180°, HGK and GKL are supplementary angles.

Because 41° + 49° = 90°, FGK and GKL are complementary angles.

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 AnglesVertical 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, m1 + m2 = 180°. (also, Angle Addition Postulate)2 and 3 form a linear pair also, so m2 + m3 = 180°. Subtracting m2 from both sides of both equations, we get m1 = 180° − m2 = m3. 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.

1. Vertical angles _____ have a common vertex.2. Two right angles are _____ complementary.3. Right angles are _____ vertical angles.4. Angles A, B, and C are _____ complementary.5. Vertical angles _____ have a common supplement.

Practice Quiz

Complete with always, sometimes, or never.

1. Vertical angles always have a common vertex.2. Two right angles are never complementary.3. Right angles are sometimes vertical angles.4. Angles A, B, and C are never complementary.5. Vertical angles always have a common supplement.

Homework AssignmentHomework Assignment

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