Sections 3 and 4

An angle is a figure formed by two rays, or sides, with a common endpoint called the vertex (plural: vertices). You can name an angle several ways: by its vertex, by a point on each ray and the vertex, or by a number.

The set of all points between the sides of the angle is the interior of an angle. The exterior of an angle is the set of all points outside the angle. Angle Name R, SRT, TRS, or 1 You cannot name an angle just by its vertex if the point is the vertex of more than one angle. In this case, you must use all three points to name the angle, and the middle point is always the vertex.

Example 1: Naming Angles A surveyor recorded the angles formed by a transit (point A) and three distant points, B, C, and D. Name three of the angles. Possible answer: BAC CAD BAD

Check It Out! Example 1 Write the different ways you can name the angles in the diagram. RTQ, T, STR, 1, 2

The measure of an angle is usually given in degrees. Since there are 360° in a circle, one degree is of a circle. When you use a protractor to measure angles, you are applying the following postulate.

Find the measure of each angle. Then classify each as acute, right, or obtuse. Example 2: Measuring and Classifying Angles A. WXV B. ZXW mWXV = 30° WXV is acute. mZXW = |130° - 30°| = 100° ZXW = is obtuse.

Check It Out! Example 2 Use the diagram to find the measure of each angle. Then classify each as acute, right, or obtuse. a. BOA b. DOB c. EOC mBOA = 40° mDOB = 125° mEOC = 105° BOA is acute. DOB is obtuse. EOC is obtuse.

Congruent angles are angles that have the same measure. In the diagram, mABC = mDEF, so you can write ABC  DEF. This is read as “angle ABC is congruent to angle DEF.” Arc marks are used to show that the two angles are congruent.

mDEG = 115°, and mDEF = 48°. Find mFEG Example 3: Using the Angle Addition Postulate mDEG = mDEF + mFEG 115 = 48 + mFEG 67 = mFEG  Add. Post. Substitute the given values. Subtract 48 from both sides. Simplify. –48°

An angle bisector is a ray that divides an angle into two congruent angles. JK bisects LJM; thus LJK  KJM.

Example 4: Finding the Measure of an Angle KM bisects JKL, mJKM = (4x + 6)°, and mMKL = (7x – 12)°. Find mJKM.

Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. Example 1A: Identifying Angle Pairs  AEB and  BED  AEB and  BED have a common vertex, E, a common side, EB, and no common interior points. Their noncommon sides, EA and ED, are opposite rays. Therefore,  AEB and  BED are adjacent angles and form a linear pair.

Check It Out! Example 1a  5 and  6 Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent.  5 and  6 are adjacent angles. Their noncommon sides, EA and ED, are opposite rays, so  5 and  6 also form a linear pair.

You can find the complement of an angle that measures x° by subtracting its measure from 90°, or (90 – x)°. You can find the supplement of an angle that measures x° by subtracting its measure from 180°, or (180 – x)°.

Check It Out! Example 2 a. complement of  E Find the measure of each of the following. b. supplement of  F = (102 – 7x)° 180  – 116.5° = 90° – (7x – 12)° = 90° – 7x° + 12° (90 – x)° (180 – x) 

An angle is 10° more than 3 times the measure of its complement. Find the measure of the complement. Example 3: Using Complements and Supplements to Solve Problems x = 3(90 – x) + 10 x = 270 – 3x + 10 x = 280 – 3x 4x = 280 x = 70 The measure of the complement,  B, is (90 – 70)  = 20 . Substitute x for m  A and 90 – x for m  B. Distrib. Prop. Divide both sides by 4. Combine like terms. Simplify. Step 1 Let m  A = x°. Then  B, its complement measures (90 – x)°. Step 2 Write and solve an equation.

Light passing through a fiber optic cable reflects off the walls of the cable in such a way that  1 ≅  2,  1 and  3 are complementary, and  2 and  4 are complementary. If m  1 = 47°, find m  2, m  3, and m  4. Example 4: Problem-Solving Application

Another angle pair relationship exists between two angles whose sides form two pairs of opposite rays. Vertical angles are two nonadjacent angles formed by two intersecting lines.  1 and  3 are vertical angles, as are  2 and  4.