a. Identify adjacent, vertical, complementary, and supplementary angles.

1. Objectives a. Identify adjacent, vertical, complementary, and supplementary angles. b. Find measures of pairs of angles. c. Find the co-ordinates of the midpoint of a line. d. Find the distance between two points

2. Vocabulary adjacent angles linear pair complementary angles supplementary angles vertical angles Mid Point Formula Distance Formula Pythagorean Theorem

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

5. AEB and BEC have a common vertex, E, • a common side, EB • no common interior points. • AEB and BEC are only adjacent angles. Example 1B: Identifying Angle Pairs Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. AEB and BEC

6. Example 1C: Identifying Angle Pairs Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. DEC and AEB • DEC and AEB share E • do not have a common side • DEC and AEB are not adjacent angles.

8. 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)°.

9. Example 1: Finding the Measures of Complements and Supplements Find the measure of each of the following. A. complement of F (90– x) 90 –59=31 B. supplement of G (180– x) 180– (7x+10)= 180 – 7x– 10 = (170– 7x)

10. Practice 1 mXYZ = 2x° and mPQR = (8x - 20)°. 1. If XYZ and PQR are supplementary, find the measure of each angle. 2. If XYZ and PQR are complementary, find the measure of each angle. 40°; 140° 22°; 68°

11. Practice 2 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. m2 = 47°, m3 = 43°, and m4 =43°.

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

13. Practice 1 Name the pairs of vertical angles. HML and JMK are vertical angles. HMJ and LMK are vertical angles. Check mHML  mJMK  60°. mHMJ  mLMK  120°.

15. Example 1: Finding the Coordinates of a Midpoint Find the coordinates of the midpoint of PQ with endpoints P(–8, 3) and Q(–2, 7). = (–5, 5)

16. Step 2 Use the Midpoint Formula: Example 2: Finding the Coordinates of an Endpoint M is the midpoint of XY. X has coordinates (2, 7) and M has coordinates (6, 1). Find the coordinates of Y. Step 1 Let the coordinates of Y equal (x, y).

17. – 2 – 7 –2 –7 Example 2 Continued Step 3 Find the x-coordinate. Set the coordinates equal. Multiply both sides by 2. 12 = 2 + x Simplify. 2 = 7 + y Subtract. –5 = y 10 = x Simplify. The coordinates of Y are (10, –5).

18. S is the midpoint of RT. R has coordinates (–6, –1), and S has coordinates (–1, 1). Find the coordinates of T. Practice 1 The coordinates of T are (4, 3).