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Angle Pair Relationships Angle Pair Relationship Essential Questions How are special angle pairs identified? 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. ____________. 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 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 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 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 ____ 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 and only if (iff): 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 and Supplementary Angles Two angles are complementary if and only if (iff) 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 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. 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 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. Two angles are supplementary if and only if (iff) 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 Complementary and 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 Congruent Angles measure Recall that congruent segments have the same ________. Congruent angles _______________ also have the same measure. Congruent Angles Two angles are congruent iff, 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 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 iff 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°. 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°