2.  TEKS Focus:  (6)(A) Verify theorems about angles formed by the intersection of lines and line segments, including vertical angles, and angle formed by parallel lines cut by a transversal and prove equidistance between the endpoints of a segment and points on its perpendicular bisector and apply these relationships to solve problems.  (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.  (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas.

3. This means that the angles do not overlap.

4.  What are Vertical Angles? What are Vertical Angles?

5. Another angle pair relationship exists between two angles whose sides form two pairs of opposite rays. Vertical angles are two nonadjacent angles whose sides are opposite rays. Remember that opposite rays will form a line.  1 and  3 are vertical angles, as are  2 and  4.

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

7.  What are Complementary Angles? What are Complementary Angles?  What are Supplementary Angles? What are Supplementary Angles?

8. Complementary and Supplementary angles do not have to be next to each other (see above), but they can be adjacent.

9. Answer always, sometimes, or never for the following statements: 1.Complementary angles are adjacent. 2.Supplementary angles form linear pairs. 3.Linear pairs form supplementary angles. Answers: 1 is sometimes; 2 is sometimes; 3 is always.

12. Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent.  AEB and  BED Example: 1 Adjacent and Linear Pair  AEB and  BEC Adjacent Only  DEC and  AEB Not Adjacent

13. Name the pairs of vertical angles.  HML and  JMK are vertical angles.  HMJ and  LMK are vertical angles. If m  HML = 60°, then m  LMK = 120°, then m  KMJ = 60°, and m  JMK = 120°. Example: 2 What do you notice about the measures of vertical angles? They are equal, therefore the vertical angles are congruent.

14. Find the measure of each of the following. A. complement of  F B. supplement of  G 90  – 59  = 31  (180 – x)  180 – (7x+10)  = 180  – 7x – 10 = (170 – 7x)  (90 – x)  Example: 3

15. 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)  Example: 4

16. Find the m  EDG and m  FDH. Are they a pair of vertical angles?  EDG and  FDH are vertical angles and they have the same measure. Example: 5 (x + 24)  (4x - 36)  4x - 36 = x + 24 3x - 36 = 24 3x = 60 x = 20 Vertical angles are congruent. Subtraction Property of Equality. Addition Property of Equality. Division Property of Equality. m  EDG = 4(20) – 36 = 44  m  FDH = 20 + 24 = 44  Substitution Property of Equality

17.  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 = (3x + 4)° and m  3 = (2x + 21) °, find m  1, m  2, m  3, and m  4.

18.  If  1   2, then m  1 = m  2.  If  3 and  1 are complementary, then  m  1 + m  3 = 90°.  (3x + 4) + (2x + 21) = 90  5x + 25 = 90  5x = 65  x = 13  m  1 = 3(13) + 4 = 43°  m  3 = 2(13) + 21 = 47°  m  2 = m  1 = 43°  m  4 = m  3 = 47°

19.  Discuss the following question within your group:  Would this example be worked the same way if the directions had said that KM was an angle bisector for  JKL?  Write on KM that it is called an angle bisector for this example.

20.  EXTRA EXAMPLES NOT USED IN COMPOSITION BOOK FOLLOW

21.  5 and  6 Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent. Example: 7 Adjacent and Linear Pair  7 and  SPU Not Adjacent  7 and  8 Not Adjacent

22. What if...? Suppose m  3 = 27.6°. Find m  1, m  2, and m  4. Example: 8

23. If  1   2, then m  1 = m  2. If  3 and  1 are complementary, then m  1 = (90 – 27.6)°. If  4 and  2 are complementary, then m  4 = (90 – 27.6)°.