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Sec 2.8: Proving Angle Relationships Description of the lesson: This section is about proving angle relationships using different theorems and postulates through two column proofs. Subject Area: Geometry Approximate amount of time to be spent on this lesson: 2 periods National or District Standards to be addressed: Use visualization, spatial reasoning, and geometric modeling to solve problems. Resources used in this lesson: Geometry Book Learning objectives: ο· ο· Students will create one proof involving supplementary and complementary angles Students will create one proof involving congruent and right angles Motivational Strategies: Start off by looking at a pair of scissors. Have the students notice that when you open the pair of scissors the blades make an angle, and one of the blades and one of the handles make a linear pair. Explain angles are everywhere. Lesson Procedures 1) Opening lecture a. Start discussing the angles in the pair of scissors. b. Tell students that todayβs lesson will be about angles c. Postulate 2.11 i. If R is in the interior of β πππ, then πβ πππ + πβ π ππ = πβ πππ. ii. If πβ πππ + πβ π ππ = πβ πππ, then R is in the interior of β πππ. d. Where have we seen this postulate before? i. Segment addition postulate e. Ex: At four oβclock, the angle between the hour and the minute hands of the clock is 120°. If the second hand stops where it bisects the angle between the hour and minute hands, what are the measures of the angles between the minute and second hands and between the second and hour hands? i. They are both are 60° by the definition of angle bisector and angle addition postulate. f. Ex: The Grand Union Flag at the left contains several angles. If πβ π΄π΅π· = 44 and πβ π΄π΅πΆ = 88, find πβ π·π΅πΆ. i. πβ π΄π΅π· + πβ π·π΅πΆ = πβ π΄π΅πΆ ii. 44 + πβ π·π΅πΆ = 88 iii. πβ π΄π΅πΆ = 44 g. Theorems i. Supplement Theorem 1. If two angles form a linear pair, then they are supplementary angles. ii. Complement Theorem 1. If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles. h. Ex. If πβ 1 and πβ 2 form a linear pair and πβ 2 = 67, find πβ 1. i. πβ 1 + πβ 2 = 180 ii. πβ 1 + 67 = 180 iii. πβ 1 = 113 i. Student Exercise i. Have students work on βCheck Your Progressβ problems 2A and 2B 1. Each row works on 2A or 2B a. Every other row 2. Have students discuss their solutions with another person from another row who worked on the other problem. j. Theorem 2.5 i. Congruence of angles in reflexive, symmetric, and transitive. 1. Reflexive a. β 1 β β 1 2. Symmetric a. If β 1 β β 2, then β 2 β β 1. 3. Transitive a. If β 1 β β 2, and β 2 β β 3, then β 1 β β 3. k. Ex. Prove the symmetric property of congruence i. Given: β π΄ β β π΅ Prove: β π΅ β β π΄ Statements Reasons Given β π΄ β β π΅ Definition of Congruence πβ π΄ = πβ π΅ Symmetric Property πβ π΅ = πβ π΄ Definition of Congruence β π΄ β β π΅ l. Theorems i. 2.6 - Angles supplementary to the same angle or to congruent angles are congruent. 1. If πβ 1 + πβ 2 = 180 and πβ 2 + πβ 3 = 180, then β 1 β β 3. ii. 2.7- Angles complementary to the same angle or to congruent angles are congruent. 1. If πβ 1 + πβ 2 = 90 and πβ 2 + πβ 3 = 90, then β 1 β β 3. m. Ex. Prove Theorem 2.7 i. Given: β 1 and β 3 are complementary. β 2 and β 3 are complementary. Prove: β 1 β β 2 Statements Reasons Given β 1 and β 3 are complementary. β 2 and β 3 are complementary. Definition of complementary angles πβ 1 + πβ 3 = 90 πβ 2 + πβ 3 = 90 Substitution πβ 1 + πβ 2 = πβ 2 + πβ 3 Reflexive Property πβ 3 = πβ 3 Subtraction Property πβ 1 = πβ 2 Definition of Congruent Angles β 1 β β 2 n. Ex. Using supplementary angles i. Given: β 1 and β 2 form a linear pair. β 2 and β 3 form a linear pair. Prove: β 1 β β 3 Statements β 1 and β 2 form a linear pair. β 2 and β 3 form a linear pair. β 1 and β 2 form a supplementary. β 2 and β 3 form a supplementary. β 1 β β 3 Reasons Given Supplement Theorem Theorem 2.3 Theorem 2.6 o. Theorems i. 2.8 If two angles are vertical angles, then they are congruent. p. Student Exercise i. If β 1 and β 2 are vertical angles and πβ = π₯ and πβ 2 = 228 β 3π₯, find πβ 1 and πβ 2. ii. Use a two column proof to solve this problem 1. Use what you know. q. Theorems i. 2.9 Perpendicular lines intersect to form four right angles ii. 2.10 All right angles are congruent iii. 2.11 Perpendicular lines form congruent adjacent angles. iv. 2.12If two angles are congruent and supplementary, then each angle is a right angle v. 2.13 If two congruent angles form a linear pair, then they are right angles. 2) Closure a. Discuss what we did for this lesson i. What were some of our main ideas of this lesson 1. Congruence Properties for angles 2. Supplementary and Complementary angles ii. Whatβs next? 1. Review day for Chapter 2 test 2. Chapter 2 Test Adaptations needed to meet studentβs needs: N/A Assessment of Objectives: ο· ο· ο· Completion of the proof for the symmetric property Completion of the vertical angles proof Participation in lecture Reflection on lesson Homework after lesson: Day 1: 5-13, 35, 37 Day 2: 14-16, 30, 32-36