1.5 Describe Angle Pair Relationships Objective: use special angle relationships to find angle measures Construction activity

Relationships based on Measurements Complementary Supplementary Adjacent

Complementary Angles Two angles that have a sum of 90 degrees. They may or may not be right next to each other.

Supplementary Angles Two angles that have a sum of 180 degrees. They may or may not be right next to each other.

Complementary or Supplementary?

Tricks to Keep From Mixing Them Up C comes first in the alphabet and 90 is smaller that 180 C for corner (90 degrees), S for straight (180 degrees) Others?

Adjacent 2 angles next to each other Must share a common side and vertex

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 1 – 2 pg.35

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. FGK and GKL, HGK and GKL, FGK and HGK ANSWER

GUIDED PRACTICE for Example 1 Are KGH and LKG adjacent angles ? Are FGK and FGH adjacent angles? Explain. 2. No, they do not share a common vertex. No, they have common interior points. ANSWER

EXAMPLE 2 Find measures of a complement and a supplement Given that 1 is a complement of 2 and m 1 = 68°, find m 2. SOLUTION a. You can draw a diagram with complementary adjacent angles to illustrate the relationship. m 2 = 90° – m 1 = 90° – 68° = 22°

EXAMPLE 2 Find measures of a complement and a supplement b. Given that 3 is a supplement of 4 and m 4 = 56°, find m 3. SOLUTION b. You can draw a diagram with supplementary adjacent angles to illustrate the relationship. m 3 = 180° – m 4 = 180° –56° = 124°

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.

Extra Example 3 Two roads intersect to form supplementary angels, angle XYW and angle WYZ. Find the measure of angle XYW and angle WYZ.

Guided Practice 3 - 5

GUIDED PRACTICE for Examples 2 and 3 3. Given that 1 is a complement of 2 and m 2 = 8o, find m 1. 82o ANSWER 4. Given that 3 is a supplement of 4 and m 3 = 117o, find m 4. 63o ANSWER 5. LMN and PQR are complementary angles. Find the measures of the angles if m LMN = (4x – 2)o and m PQR = (9x + 1)o. ANSWER 26o, 64o

Angle Pairs Linear Pair Vertical Angles

Linear Pair Adjacent angles that form a straight line Their noncommon sides form opposite rays.

Questions True or False… Every linear pair of angles is supplementary. Every pair of supplementary angles is a linear pair.

Vertical Angles Sides form opposite rays

EXAMPLE 4 Identify angle pairs Identify all of the linear pairs and all of the vertical angles in the figure at the right. SOLUTION To find vertical angles, look or angles formed by intersecting lines. 1 and 5 are vertical angles. ANSWER To find linear pairs, look for adjacent angles whose noncommon sides are opposite rays. 1 and 4 are a linear pair. 4 and 5 are also a linear pair. ANSWER

EXAMPLE 5 Find angle measures in a linear pair Two angles form a linear pair. The measure of one angle is 5 times the measure of the other. Find the measure of each angle. ALGEBRA SOLUTION Let x° be the measure of one angle. The measure of the other angle is 5x°. Then use the fact that the angles of a linear pair are supplementary to write an equation.

Find angle measures in a linear pair EXAMPLE 5 Find angle measures in a linear pair xo + 5xo = 180o Write an equation. 6x = 180 Combine like terms. x = 30o Divide each side by 6. The measures of the angles are 30o and 5(30)o = 150o. ANSWER

GUIDED PRACTICE For Examples 4 and 5 Do any of the numbered angles in the diagram below form a linear pair? Which angles are vertical angles? Explain. 6. ANSWER No, no adjacent angles have their noncommon sides as opposite rays, 1 and 4 , 2 and 5, 3 and 6, these pairs of angles have sides that from two pairs of opposite rays.

GUIDED PRACTICE For Examples 4 and 5 7. The measure of an angle is twice the measure of its complement. Find the measure of each angle. ANSWER 60°, 30°

Review Describe each with a short sentence or sketch. Complementary Angles Supplementary Angles Adjacent Angles Linear Pair Vertical Angles

Homework 1, 4 -16 evens, 17 – 28, 31-38, 49 – 53 Quiz coming up over 1.4-1.5 (study bottom of page 41)