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Angles – sides and vertex There is another case where two rays can have a common endpoint. angle This figure is called an _____. Some parts of angles have special names. S 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. R vertex side T Naming Angles There are several ways to name this angle. S 1) Use the vertex and a point from each side. SRT or TRS The vertex letter is always in the middle. 2) Use the vertex only. R 1 side R vertex If there is only one angle at a vertex, then the angle can be named with that vertex. 3) Use a number. 1 T Angles An angle is a figure formed by two noncollinear rays that have a common endpoint. Symbols: D Definition of Angle DEF FED E 2 E F 2 Angles 1) Name the angle in four ways. C ABC A CBA 1 B 1 B 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 obtuse angle 90 < m A < 180 A A right angle m A = 90 acute angle 0 < m A < 90 Angle Measure Classify each angle as acute, obtuse, or right. 110° 40° 90° Obtuse Right Acute 50° 130° Acute Obtuse 75° Acute Straight Angles Opposite rays are two rays that are part of a the same line and have ___________ only their endpoints in common. Y X Z opposite rays XY and XZ are ____________. The figure formed by opposite rays is also referred to as a straight angle A straight angle measures 180 degrees. ____________. Congruent Angles measure Recall that congruent segments have the same ________. Congruent angles _______________ also have the same measure. Congruent Angles Two angles are congruent if they have the same degree measure ______________. Definition of Congruent Angles B V iff 50° 50° B V mB = mV Congruent Angles arcs To show that 1 is congruent to 2, we use ____. 1 2 To show that there is a second set of congruent angles, X and Z, we use double arcs. This “arc” notation states that: X Z X mX = mZ Z Angles 1) Name all angles having W as their vertex. X 1 2 W 1 2 XWZ Y 2) What are other names for XWY or 1 ? YWX 3) Is there an angle that can be named No! Z W? 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 _____________ adjacent angles adjacent = next to, joining. A B 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. C BD ____ 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 Definition of Adjacent Angles J R 1 and 2 are adjacent with the same vertex R and 2 1 common side RM N Adjacent Angles Determine whether 1 and 2 are adjacent angles. No. They have a common vertex B, but no common side _____________ 2 1 B 1 Yes. They have the same vertex G and a common side with no interior points in common. 2 G N L J 2 1 No. They do not have a common vertex or a____________ common side LN The side of 1 is ____ JN The side of 2 is ____ Vertical Angles When two lines intersect, four ____ angles are formed. There are two pair of nonadjacent angles. vertical angles These pairs are called _____________. 4 1 3 2 Vertical Angles Two angles are vertical if they are two nonadjacent angles formed by a pair of intersecting lines. Vertical angles: Definition of Vertical Angles 4 1 3 1 and 3 2 2 and 4 Vertical Angles Vertical angles are congruent. Theorem 3-1 Vertical Angle Theorem n 2 m 1 3 3 1 4 2 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 X 2 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 Two angles form a linear pair if: A) they are adjacent and B) their noncommon sides are opposite rays A Definition of Linear Pairs D B 1 2 1 and 2 are a linear pair. BA and BD form AD 1 2 180 Linear Pairs of Angles In the figure, CM and CE are opposite rays. 1) Name the angle that forms a linear pair with 1. ACE H T A 2 1 ACE and 1 have a common side CA the same vertex C, and opposite rays 3 4 C M CM and CE 2) Do 3 and TCM form a linear pair? Justify your answer. No. Their noncommon sides are not opposite rays. E Complementary Angles Two angles are complementary if the sum of their degree measure is 90. E D A Definition of Complementary Angles B 30° 60° F C mABC + mDEF = 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. E A B D 30° C 60° F 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° 15° H P mH + mI = 90 Q 40° mPHQ + mQHS = 90 50° H S U T I 60° V mTZU + mVZW = 90 30° Z W 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 the sum of their degree measure is 180. D C Definition of Supplementary Angles 50° A 130° B E mABC + mDEF = 50 + 130 = 180 F Supplementary Angles Some examples of supplementary angles are shown below. H 75° 105° I mH + mI = 180 Q 130° 50° H P S U V 60° 120° 60° Z T mPHQ + mQHS = 180 mTZU + mUZV = 180 and W mTZU + mVZW = 180 Vertical Angles Find the value of x in the figure: (x – 10)° 125° The angles are vertical angles. (x – 10) = 125. x – 10 = 125. x = 135. Congruent Angles Suppose A B and mA = 52. Find the measure of an angle that is supplementary to B. A B 52° B + 1 = 180 1 = 180 – B 1 = 180 – 52 1 = 128° 1 Congruent Angles G D 1 1) If m1 = 2x + 3 and the m3 = 3x + 2, then find the m3 x = 17; 3 = 37° A 4 3 2 B C E H 2) If mABD = 4x + 5 and the mDBC = 2x + 1, then find the mEBC x = 29; EBC = 121° 3) If m1 = 4x - 13 and the m3 = 2x + 19, then find the m4 x = 16; 4 = 39° 4) If mEBG = 7x + 11 and the mEBH = 2x + 7, then find the m1 x = 18; 1 = 43°