Warm up 9/15/14 Answer the question and draw a picture if you can: 1) What is a right angle? 2) What is an acute angle? 3) What is an obtuse angle? 4) What is a vertical angle?

Angle Pair Relationships

Angle Pair Relationship Essential Questions How do I prove geometric theorems involving lines, angles? Standard: MCC9-12.G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

Straight Angles Opposite rays ___________ are two rays that are part of 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

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

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

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 ____

Linear Pairs of Angles Note: Definition of Linear Pairs Two angles form a linear pair if and only if (iff): 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.

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 iff they are two 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 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°.

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.

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

Complementary and Supplementary 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

Complementary and 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 and only if (iff) the sum of their degree measure is 180. 130° D E F 50° A B C mABC + mDEF = 50 + 130 = 180

Complementary and 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

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°