Angle Pair RelationshipsSection 1.5

VocabularyAn angle consists of two different rays with the same endpoint.

The rays are the sides of the angle.

The endpoint is the vertex of the angle.

VocabularyAn angle is named with three points, just the vertex, or a number.

An angle can be named with just the vertex only if it is the only angle with that particular vertex.

The vertex is always the middle point when naming an angle with three points.

VocabularyWhen naming an angle, always remember to put a symbol in front. Otherwise you are naming a point or plane.

To denote the measure of an angle, we write an “m” in front of the angle sign: o

VocabularyAn acute angle has a measure between 0o and 90o

A right angle has a measure of 90o

An obtuse angle has a measure between 90o and 180o

A straight angle has a measure of 180o

VocabularyTwo angle are congruent angles if they have the same measure.

An angle bisector is a ray that divides an angle into two congruent angles.

VocabularyTwo angles are complementary angles if the sum of their measures is 90o.

Two angles are supplementary angles if the sum of their measures is 180o.

VocabularyAdjacent angles are two angles that share a common vertex and side, but have no common interior points.

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.

EXAMPLE 2 Find measures of a complement and a supplement

b. You can draw a diagram with supplementary adjacent angles to illustrate the relationship. m 3 = 180° – m 4 = 180° –56° = 124°

SOLUTION

b. Given that 3 is a supplement of 4 and m 4 = 56°, find m 3.

GUIDED PRACTICE for Examples 2 and 3

5. LMN and PQR are complementary angles. Find the measures of the angles if m LMN = (4x – 2)° and m PQR = (9x + 1)°.

VocabularyTwo adjacent angles are a linear pair if their noncommon sides are opposite rays.

The angles in a linear pair are supplementary angles.

VocabularyTwo angles are vertical angles if their sides form two pairs of opposite rays.

Vertical angles are congruent to each other.

SOLUTION

EXAMPLE 4 Identify angle pairs

To find vertical angles, look or angles formed by intersecting lines.

To find linear pairs, look for adjacent angles whose noncommon sides are opposite rays.

Identify all of the linear pairs and all of the vertical angles in the figure at the right.

1 and 5 are vertical angles.ANSWER

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

Angle Addition PostulateIf P is in the interior of , then the measure of is equal to the sum of the measures of and .

VocabularyTo show that two angles in a diagram are congruent, we put an arc inside each angle.

Assignmentp. 38: 3-27(odds)