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Solving Linear Pairs and Angle Relationships in Geometry**

This review covers key concepts in geometry focused on solving problems related to linear pairs, complementary and supplementary angles, and systems of equations. Students will learn to find angle measures using algebraic expressions and solve equations involving angles formed by intersecting lines. Additionally, various theorems and properties of angles, such as vertical angles and angle bisectors, are addressed. Step-by-step solutions are provided to guide students through the reasoning process while reinforcing their understanding of geometric relationships.

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Solving Linear Pairs and Angle Relationships in Geometry**

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  1. Review for Unit 1 Test

  2. 1. 1 and 2 are a linear pair.1 = 5x + 252 = 4x + 20 Find m2 5x + 25 + 4x + 20 = 180 9x + 45 = 180 9x = 135 x = 15 2= 4(15) + 20 = 80

  3. x + 4y – 15 + 2x + y – 10 = 6x – 5y -3x + 10y = 25 M Y is the midpoint of RL mEYL = (6x – 5y)Write and solve a system of equations to find the values of x and y. 2. YI bisects LYE R x + 4y – 15 = 2x + y – 10 -x + 3y = 5 Y L 2(x + 4y – 15) = 6x – 5y 2x + 8y – 30 = 6x – 5y -4x + 13y = 30 I E (2x + y – 10) (x + 4y – 15) 2(2x + y – 10) = 6x – 5y 4x + 2y – 20 = 6x – 5y -2x + 7y = 20 x = 25 y = 10

  4. 3. Six times an angle’s supplement is 5 less than the angle. Find the measure of the angle’s supplement. 6(180 – x) = x – 5 1080 – 6x = x – 5 1085 = 7x 155 = x Supplement = 180 – 155 = 25

  5. 4. 1 and 2 form a right angle.1 = 10x + 122 = 3x Find m1 10x + 12 + 3x = 90 13x + 12 = 90 13x = 78 x = 6 1= 10(6) + 12= 72

  6. 5. 1 and 2 are complementary.1 is 15 more than twice 2 Find m2 2 = x; 1 = 90 – x 90 – x = 2x + 15 75 = 3x 25 = x

  7. 6. 1 and 2 are vertical angles.1 = 3x2 – 2x + 62 = x2 – 11x + 2 Find the value of x. 3x2 – 2x + 6 = x2 – 11x + 2 2x2 + 9x + 4 = 0 (2x + 1)(x + 4) = 0 x = -½ x = -4 (both work!)

  8. 7. The sum of an angle’s supplement and its complement is equal to 8 times the angle. Find the measure of the angle. 180 – x + 90 – x = 8x 270 – 2x = 8x 270 = 10x 27 = x

  9. 8. JM intersects GW at point J GJM = 2x + 5MJW = x + 25Draw and label a diagram. G J 2x + 5 W x + 25 M

  10. M Y is the midpoint of RL 9. Can you assume: YI bisects LYE R Y L I E d) R, Y, L are collinear e) Angle LYI = angle EYI a) SO + OU = SU c) Angle LYE is right b) RY = YL f) MY = YI e) Angle LYI = angle MYR d) yes e) yes, b/c you’re told BISECTOR a) yes c) no! b) yes, b/c you’re told MIDPOINT f) no! e) yes, b/c they are vertical angles

  11. 10. Anangle is twice it’s complement. Find the measure of the angle’s supplement. x = 2(90 – x) x = 180 – 2x 3x = 180 x = 60 Supplement = 180 – 60 = 120

  12. 11. Solve the system using either substitution or elimination. 3x + 2y = 11 -6x + y = -32 x = 5 y = -2

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