Angles – sides and vertex There is another case where two rays can have a common endpoint. angle This figure is called an _____. S Some parts of angles have special names. side vertex The common endpoint is called the ______, and the two rays that make up the sides of the angle are called the sides of the angle. T side R vertex

Naming Angles There are several ways to name this angle. 1) Use the vertex and a point from each side. S SRT or TRS side The vertex letter is always in the middle. 2) Use the vertex only. 1 R T side vertex R If there is only one angle at a vertex, then the angle can be named with that vertex. 3) Use a number. 1

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

Angles 1) Name the angle in four ways. ABC CBA B 1 2) Identify the vertex and sides of this angle. vertex: Point B sides: BA and BC

Angle Measure Once the measure of an angle is known, the angle can be classified as one of three types of angles. These types are defined in relation to a right angle. Types of Angles A A A obtuse angle 90 < m A < 180 right angle m A = 90 acute angle 0 < m A < 90

Angle Measure Classify each angle as acute, obtuse, or right. Obtuse 110° 90° 40° Obtuse Right Acute 75° 50° 130° Acute Obtuse Acute

Straight Angles Opposite rays ___________ are two rays that are part of a the same line and have only their endpoints in common. X Y Z opposite rays XY and XZ are ____________. The figure formed by opposite rays is also referred to as a ____________. A straight angle measures 180 degrees. straight angle

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

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

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

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

Angle Pair Relationships

Angle Pair Relationship Essential Questions How are special angle pairs identified?

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

Adjacent Angles Adjacent angles are angles that: Definition of Adjacent Angles Adjacent angles are angles that: A) share a common side B) have the same vertex, and C) have no interior points in common M J N R 1 2 1 and 2 are adjacent with the same vertex R and common side

Adjacent Angles Determine whether 1 and 2 are adjacent angles. No. They have a common vertex B, but _____________ 1 2 B no common side Yes. They have the same vertex G and a common side with no interior points in common. 1 2 G 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 ____

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

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

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

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

Adjacent Angles and Linear Pairs of Angles Determine whether 1 and 2 are adjacent angles. No. 1 2 Yes. 1 2 X D Z In this example, the noncommon sides of the adjacent angles form a ___________. straight line linear pair These angles are called a _________

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

Linear Pairs of Angles 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.

Complementary Angles Two angles are complementary if Definition of Complementary Angles Two angles are complementary if the sum of their degree measure is 90. 60° D E F 30° A B C mABC + mDEF = 30 + 60 = 90

Complementary Angles 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. 60° D E F 30° A B C Complementary angles DO NOT need to have a common side or even the same vertex.

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

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

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

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

Congruent Angles Suppose A  B and mA = 52. Find the measure of an angle that is supplementary to B. A 52° B 52° 1 B + 1 = 180 1 = 180 – B 1 = 180 – 52 1 = 128°

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