1. ANGLE PAIR RELATIONSHIPS

2. Definition of Angle An angle is a figure formed by two noncollinear rays that have a common endpoint. E D F 2 Symbols: DEF 2 E FED

3. B A 1 C 1) Name the angle in four ways. ABC 1 B CBA 2) Identify the vertex and sides of this angle. Point B BA andBC vertex: sides:

4. W Y X 1) Name all angles having W as their vertex. 1 2 Z 1 2 2) What are other names for ? 1 XWY or YWX 3) Is there an angle that can be named ? W No! XWZ

5. When you “split” an angle, you create two angles. D A C B 1 2 The two angles are called _____________ adjacent angles  1 and  2 are examples of adjacent angles. They share a common ray. Name the ray that  1 and  2 have in common. ____ adjacent = next to, joining.

6. Determine whether  1 and  2 are adjacent angles. No. They have a common vertex B, but _____________ no common side 1 2 B 1 2 G Yes. They have the same vertex G and a common side with no interior points in common. N 1 2 J L No. They do not have a common vertex or ____________ a common side The side of  1 is ____ The side of  2 is ____

7. Definition of Linear Pairs Two angles form a linear pair if and only if (iff):  1 and  2 are a linear pair. A) they are adjacent and B) their noncommon sides are opposite rays C A D B 1 2

8. In the figure, and are opposite rays. 1 2 M 4 3 E H T A C 1) Name the angle that forms a linear pair with  1.  ACE  ACE and  1 have a common side the same vertex C, and opposite rays and 2) Do  3 and  TCM form a linear pair? Justify your answer. No. Their noncommon sides are not opposite rays.

9. Definition of Complementary Angles 30° A B C 60° D E F Two angles are complementary if and only if (iff) The sum of their degree measure is 90. m  ABC + m  DEF = 30 + 60 = 90 Notes: “m” stands for “measure”

10. 30° A B C 60° D E F If two angles are complementary, each angle is a complement of the other.  ABC is the complement of  DEF and  DEF is the complement of  ABC. Complementary angles DO NOT need to have a common side or even the same vertex.

11. 15° H 75° I Some examples of complementary angles are shown below. m  H + m  I = 90 m  PHQ + m  QHS = 90 50° H 40° Q P S 30° 60° T U V W Z m  TZU + m  VZW = 90

12. Definition of Supplementary Angles If the sum of the measure of two angles is 180, they form a special pair of angles called supplementary angles. Two angles are supplementary if and only if (iff) the sum of their degree measure is 180. 50° A B C 130° D E F m  ABC + m  DEF = 50 + 130 = 180

13. 105° H 75° I Some examples of supplementary angles are shown below. m  H + m  I = 180 m  PHQ + m  QHS = 180 50° H 130° Q P S m  TZU + m  UZV = 180 60° 120° T U V W Z 60° and m  TZU + m  VZW = 180

14. Recall that congruent segments have the same ________. measure _______________ also have the same measure. Congruent angles

15. Definition of Congruent Angles Two angles are congruent iff, they have the same ______________. degree measure 50° B V  B   V iff m  B = m  V

16. 1 2 To show that  1 is congruent to  2, we use ____. arcs Z X To show that there is a second set of congruent angles,  X and  Z, we use double arcs. X  ZX  Z m  X = m  Z This “arc” notation states that:

17. 1) If m  1 = 2x + 3 and the m  3 = 3x + 2, then find the m  3 2) If m  ABD = 4x + 5 and the m  DBC = 2x + 1, then find the m  EBC 3) If m  1 = 4x - 13 and the m  3 = 2x + 19, then find the m  4 4) If m  EBG = 7x + 11 and the m  EBH = 2x + 7, then find the m  1 x = 17;  3 = 37° x = 29;  EBC = 121° x = 16;  4 = 39° x = 18;  1 = 43° A B C D E G H 1 2 3 4

18. When two lines intersect, ____ angles are formed. four 1 2 3 4 There are two pair of nonadjacent angles. These pairs are called _____________. vertical angles

19. Definition of Vertical Angles Two angles are vertical iff they are two nonadjacent angles formed by a pair of intersecting lines. 1 2 3 4 Vertical angles:  1 and  3  2 and  4

20. Theorem 3-1 Vertical Angle Theorem Vertical angles are congruent. 1 4 3 2 m n  1   3  2   4

21. Find the value of x in the figure: The angles are vertical angles. So, the value of x is 130°. 130° x°

22. Find the value of x in the figure: The angles are vertical angles. (x – 10) = 125. (x – 10)° 125° x – 10 = 125. x = 135.

23. ACUTE ANGLE Less than 90 degrees

24. OBTUSE ANGLE More than 90° and less than 180°

25. RIGHT ANGLE A 90 degree angle