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1.5 Describe Angle Pair Relationships Goal Use special angle relationships to find angle measures. Your Notes VOCABULARY Complementary angles Two angles whose sum is 90° Supplementary angles Two angles whose sum is 180° Adjacent angles Two angles that share a common vertex or side but have no common interior points Linear pair Two adjacent angles are a linear pair if their noncommon sides are opposite rays. Vertical angles Two angles are vertical angles if their sides form two pairs of opposite rays. 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. In Example 1, BDE and CDE share a common vertex. But they share common __interior__ points, so they are not adjacent angles. Solution Because _52°_ + _38°_ = 90°, _ABD_ and _CDB_ are _complementary_ angles Because _52°_ + _128°_ = 180°, _ABD_ and _EDB_ are _supplementary_ angles. Because _CDB_ and _BDE_ share a common vertex and side, they are _adjacent_ angles. Your Notes Example 2 Find measures of complements and supplements a. Given that 1 is a complement of 2 and m2 = 57°, find m1. b. Given that 3 is a complement of 4 and m4 = 41°, find m3. Angles are sometimes named with numbers. An angle measure in a diagram has a degree symbol. An angle name does not. Solution a. You can draw a diagram with complementary adjacent angles to illustrate the relationship. m1 = _90°_ – m2__ = _90°_ – _57°_ = _33°_ b. You can draw a diagram with supplementary adjacent angles to illustrate the relationship. m 3 = _180°_ – _m4_= _180°_ – _41°_ = _139°_ Checkpoint Complete the following exercises. 1. In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles. Complementary: DEF and ABC, supplementary; FEG and ABC, adjacent: DEF and FEG 2. Given that 1 is a complement of 2 and m1 = 73°, find m2. 17° 3. Given that 3 is a supplement of 4 and m4 = 37°, find m3. 143° Your Notes Example 3 Find angle measures Basketball The basketball pole forms a pair of supplementary angles with the ground. Find mBCA and mDCA. In a diagram, you can assume that a line that looks straight is straight. In Example 3, B, C, and D lie on BD . So, BCD is a __straight__ angle. Solution Step 1 Use the fact that __180°__ is the sum of the measures of supplementary angles. mBCA + mDCA = _180°_ (_3x + 8_)° + (_4x – 3_)° = _180°_ _7x + 5_ = _180°_ _7x_ = _175_ _x_ = 25 Write equation. Substitute. Combine like terms. Subtract. Divide. Step 2 Evaluate the original expressions when x = _25_. mBCA = (_3x + 8_)° = (_3 • 25 + 8_)° = _83°_. mDCA = (_4x – 3_)° = (_4 • 25 – 3_)° = _97°_. The angle measures are _83°_ and _97°_. Checkpoint Complete the following exercise. 4. In Example 3, suppose the angle measures are (5x + 1)° and (6x + 3)°. Find mBCA and mDCA. 81° and 99° Your Notes Example 4 Identify angle pairs Identify all of the linear pairs and all of the vertical angles in the figure at the right. Solution In the diagram, one side of 1 and one side of 4 are opposite rays. But the angles are not a linear pair because they are not _adjacent_. To find vertical angles, look for angles formed by _intersecting lines_. _1_ and _3_ are vertical angles. To find linear pairs, look for adjacent angles whose noncommon sides are _opposite rays_. _1_ and _2_ are a linear pair. _2_ and _3_ are a linear pair. Checkpoint Complete the following exercise. 5. Identify all of the linear pairs and all of the vertical angles in the figure. linear pairs none; vertical angles: 1 and 4, 2 and 5, 3 and 6 Example 5 Find angle measures in a linear pair Two angles form a linear pair. The measure of one angle is 4 times the measure of the other. Find the measure of each angle. You may find it useful to draw a diagram to represent a word problem like the one in Example 5. Solution Let x° be the measure of one angle. The measure of the other angle is _4x°_. Then use the fact that the angles of a linear pair are _supplementary_ to write an equation. _x°_ + _4x°_ = 180° _5x_ = _180_ _x_ = _36_ Write an equation. Combine like terms. Divide each side by _5_. The measures of the angles are _36°_ and _4(36°)_ = _144°_. Your Notes Checkpoint Complete the following exercise. 6. Two angles form a linear pair. The measure of the other. Find the measure of each angle. 45° and 135° CONCEPT SUMMARY: INTERPRETING A DIAGRAM There are some things you can conclude from a diagram, and some you cannot. For example, here are some things that you can conclude from the diagram at the right. All points shown are _coplanar_. Points, A, B, and C are _collinear_, and B is between A and C. AC , BD , and BE_intersect_ at point B. DBE and EBC are _adjacent_ angles, and ABC is a _straight angle_. Point E lies in the _interior_ of DBC. In the diagram above, you cannot conclude that AB BC ,, that DBE EBC, or that ABD is a right angle. This information must be indicated, as shown at the right. Homework ________________________________________________________________________ ________________________________________________________________________