Section 1 5 Exploring Angle Pairs Objective Students

Section 1. 5 Exploring Angle Pairs

Objective: Students will be able to: • identify special angle pairs and use their relationships to find angle measures 1. 5 Exploring Angle Pairs

Vocabulary: Key Vocabulary adjacent angles complementary angles linear pair vertical angles supplementary angles angle bisector 1. 5 Exploring Angle Pairs

Special angle pairs can help you identify geometric relationships. We use them to find angle measures. 1. 5 Exploring Angle Pairs

Problem 1: Use the diagram at the right. Is the statement true? Explain a. <BFD and <CFD are adjacent angles. b. <AFB and <EFD are vertical angles c. <AFE and <BFC are complementary. 1. 5 Exploring Angle Pairs

Problem 1 Solutions: a. <BFD and <CFD are adjacent angles. No. <CDF is included (inside of) in <BFD b. <AFB and <EFD are supplementary angles? Yes. They share vertex F. c. <AFE and <BFC are complementary. Yes since 620 + 280 = 900 1. 5 Exploring Angle Pairs

Problem 1 Solutions: a. <BFD and <CFD are vertical angles? . No. <CDF do not share a vertex <BFD b. <AFB and <EFD are supplementary angles? Since <EFC = 180 and we know <EFC = 620, <BFC = 280, then <AFB = 900. Thus NO because < AFB + < EFD = 900 +1180 > 1800 c. <AFE and <BFC are adjacent angles? Yes, they share a vertex F. 1. 5 Exploring Angle Pairs

1. 5 Exploring Angle Pairs

Problem 2: What can you conclude from the information in the diagram? 1. 5 Exploring Angle Pairs

Problem 2 Solution: < 1 and < 2 are congruent. This means they have the same measurement in degrees. < 1 + < 2 are congruent to < 4. This means they have the same measurement in degrees. They are also opposites which means they are vertical angles. < 3 and < 5 are vertical and congruent. 1. 5 Exploring Angle Pairs

Problem 2 b: Can you make each conclusion from the information in the diagram? Explain. a. b. c. d. Segment TW is congruent to Segment WV Segment PW is congruent to Segment WQ <TWQ is a right angle Segment TV bisects Segment PQ 1. 5 Exploring Angle Pairs

Problem 2 b Solutions: Can you make each conclusion from the information in the diagram? Explain. a. Segment TW is congruent to Segment WV Since they have a red mark , YES they are congruent. b. Segment PW is congruent to Segment WQ We cannot conclude this. No marks are given and we do not know the length. c. <TWQ is a right angle We cannot conclude this statement since no little square is shown although it looks like it is. d. Segment TV bisects Segment PQ No. Segment PQ is bisecting segment TV. 1. 5 Exploring Angle Pairs

A linear pair is a pair of adjacent angles whose noncommon sides are opposite rays. The angles of a linear pair form a straight angle. Remember a straight line = 1800 So supplementary angles = 1800 1. 5 Exploring Angle Pairs

Problem 3: <KPL and <JPL are a linear pair, m<KPL = 2 x + 24, and m<JPL = 4 x + 36. What are the measures of <KPL and <JPL? 1. 5 Exploring Angle Pairs

Problem 3 Solution: <KPL and <JPL are a linear pair, m<KPL = 2 x + 24, and m<JPL = 4 x + 36. What are the measures of <KPL and <JPL? 1. Make a sketch (drawing) 2. Label 2 x + 24 3. Use algebra to solve. (2 x + 24) + (4 x + 36) = 180 6 x + 60 = 180 6 x = 120 x = 20 K L 4 x + 36 P J 4. Plug in the variable and find the measure of the angle. < KPL = (2(20) + 24) = 40 + 24 = 64 < JPL = (4(20) + 36) = 80 + 36 = 116 1. 5 Exploring Angle Pairs

An angle bisector is a ray that divides an angle into two congruent (same degree) angles. Its endpoint is a the angle vertex. Within the ray, a segment with the same endpoint is also an angle bisector. The ray or segment bisects the angle. In the diagram, Ray AY is the angle bisector of <XAZ, so m<XAY = m<YAZ. 1. 5 Exploring Angle Pairs

Problem 4: Ray AC bisects <DAB. If m<DAC = 58, what is m<DAB? 1. 5 Exploring Angle Pairs

Problem 4 Solution: Ray AC bisects <DAB. If m<DAC = 58, what is m<DAB? 1. Make a sketch (drawing) 2. Label 58 D 3. Use algebra to solve. Since AC bisects (cuts in half) ADB, we know that <CAB is congruent to < DCA <CDB = 2( <DAC) = 2(58) = 118 C 58 B A 1. 5 Exploring Angle Pairs

Lesson Check 1. 5 Exploring Angle Pairs