ANGLE PAIR RELATIONSHIPS
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
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:
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
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.
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 ____
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
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.
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 = = 90 Notes: “m” stands for “measure”
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.
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
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 ° A B C 130° D E F m ABC + m DEF = = 180
105° H 75° I Some examples of supplementary angles are shown below. m H + m I = 180 m PHQ + m QHS = ° H 130° Q P S m TZU + m UZV = ° 120° T U V W Z 60° and m TZU + m VZW = 180
Recall that congruent segments have the same ________. measure _______________ also have the same measure. Congruent angles
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
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 ZX Z m X = m Z This “arc” notation states that:
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
When two lines intersect, ____ angles are formed. four There are two pair of nonadjacent angles. These pairs are called _____________. vertical angles
Definition of Vertical Angles Two angles are vertical iff they are two nonadjacent angles formed by a pair of intersecting lines Vertical angles: 1 and 3 2 and 4
Theorem 3-1 Vertical Angle Theorem Vertical angles are congruent m n 1 3 2 4
Find the value of x in the figure: The angles are vertical angles. So, the value of x is 130°. 130° x°
Find the value of x in the figure: The angles are vertical angles. (x – 10) = 125. (x – 10)° 125° x – 10 = 125. x = 135.
ACUTE ANGLE Less than 90 degrees
OBTUSE ANGLE More than 90° and less than 180°
RIGHT ANGLE A 90 degree angle