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Understanding Distance, Midpoints, and Angle Relationships in Geometry

This lesson explores fundamental concepts in geometry, including the distance between two points, finding midpoints, and recognizing the properties of angles, particularly complementary and supplementary angles. We will solve problems involving the measures of angles, specifically when dealing with perpendicular angles and vertical angle relationships. Moreover, we will analyze a variety of examples to strengthen understanding, including calculations of angles based on given algebraic expressions.

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Understanding Distance, Midpoints, and Angle Relationships in Geometry

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  1. Week 3 Warm Up 08.30.11 1) What is the distance between the points? 2) What is the midpoint? ( 0, 5 ) and ( 14, 1 )

  2. 43) m∡A = 22.5, m∡B = 67.5 37) supp Homework 44) m∡A = 20, m∡B = 160 45) m∡A = 73, m∡B = 17 38) neither 46) m∡A = 14, m∡B = 76 47) m∡A = 89, m∡B = 1 39) comp 48) m∡A = 11, m∡B = 79 40) neither 49) m∡A = 129, m∡B = 51 51) m∡A = 157, m∡B = 23 52) m∡A = 77, m∡B = 103 53) m∡A = 122, m∡B = 56 41) 42)

  3. Perpendicular Angles If one angle is 90º then all four angles from two lines are 90º.

  4. Perpendicular Angles If one angle is 90º then all four angles from two lines are 90º.

  5. A E B F D C What is the measure of ∠CFE? Ex 1 (11x – 1)º (3x – 7)º complementary m∠DFE + m∠EFA =90º 3x – 7 + 11x - 1 = 90º 14x - 8 = 90 14x = 90 + 8 14x = 98 x = 7

  6. A E B F D C What is the measure of ∠CFE? Ex 1 (11x – 1)º (3x – 7)º complementary m∠DFE + m∠EFA =90º 3x – 7 + 11x - 1 = 90º 14x - 8 = 90 14x = 90 + 8 14x = 98 x = 7

  7. A E B F D C What is the measure of ∠CFE? Ex 1 (11x – 1)º (3x – 7)º x = 7 90º m∠DFE = 3x - 7 = 3( 7 ) - 7 = 21 - 7 m∠DFE = 14º m∠CFD + m∠DFE =m∠CFE 90º+ 14º = m∠CFE 104º = m∠CFE

  8. A B C D E What is the measure of ∠AEB? Ex 2 (4x – 15)º x + 20º m∠AED and m∠DEC are linear pairs m∠AED + m∠DEC =180º 4x - 15 + x + 20 = 180º 5x + 5 = 180 5x = 180 - 5 5x = 175 x = 35

  9. A B C D E What is the measure of ∠AEB? Ex 2 (4x – 15)º x + 20º m∠AED and m∠DEC are linear pairs m∠AED + m∠DEC =180º 4x - 15 + x + 20 = 180º 5x + 5 = 180 5x = 180 - 5 5x = 175 x = 35

  10. A B C D E What is the measure of ∠AEB? Ex 2 (4x – 15)º x + 20º x = 35 m∠DEC = x + 20 m∠DEC =35 + 20 m∠DEC =55º ∠DEC and ∠AEB are vertical angles m∠DEC = m∠AEB 55º = m∠AEB

  11. Review Supplementary angles and ____ ____ add up to ____º. Do: 2 What is the measure of ∠BAF? D E (x + 12)º C (3x + 2)º A F G B Handout Section 1.6 Day 3 Assignment:

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