Two-Step Equations and Angle Measures

Two-Step Equations and Angle Measures

• Supplementary angles: Two angles whose measures add up to 180 degrees.
• Complementary angles: Two angles whose measures add up to 90 degrees.
• Adjacent angles: Two angles that share a common vertex and a common side but no common interior points.
• Equation solving: Use algebra to solve for unknown variables (like x) when given relationships between angles.

Example Problems (Supplementary Angles)

• Q1: Two supplementary angles measure (7x + 10) and (5x + 20). Solve for x.

• Solution: (7x + 10) + (5x + 20) = 180 ; 12x + 30 = 180; 12x = 150; x = 12.5

• Q2: Supplementary angles measure (6x + 15) and (4x + 25). Solve for x.

• Solution: (6x + 15) + (4x + 25) = 180 ; 10x + 40 = 180 ; 10x = 140 ; x = 14

• Q3: Supplementary angles measure (5x + 20) and (3x + 10). Solve for x.

• Solution: (5x + 20) + (3x + 10) = 180 ; 8x + 30 = 180 ; 8x = 150 ; x = 18.75

Example Problems (Adjacent and Complementary Angles)

• Q4: ∠G and ∠H are adjacent, and their measures sum to 105°. Angle G measures (2x + 6), and Angle H is four times the size of Angle G. Find x.

• Solution: (2x + 6) + 4(2x + 6) = 105 ; (2x + 6) + (8x + 24) = 105 ; 10x + 30 = 105 ; 10x = 75 ; x = 7.5

• Q5: ∠C and ∠D are adjacent, and their measures sum to 120°. Angle C measures (4x + 10)°, and Angle D is twice the size of Angle C. Solve for x.

• Solution:(4x + 10) + 2(4x + 10) = 120 ; (4x + 10) + (8x + 20) = 120 ; 12x + 30 = 120 ; 12x = 90 ; x = 7.5

• Q6: Two complementary angles have measures of 8x-58 and 2x+35. Solve for x.

• Solution: (8x - 58) + (2x + 35) = 90 ; 10x - 23 = 90 ; 10x = 113 ; x = 11.3

• Q7: The measures of two complementary angles are 9x-10 and x+100. Find x.

• Solution: (9x - 10) + (x + 100) = 90 ; 10x + 90 = 90 ; 10x = 0 ; x = 0

• Q8: Complementary angles have measures of 6x + 25 and 4x − 54. Find x.

• Solution: (6x + 25) + (4x − 54) = 90; 10x - 29 = 90; 10x = 119; x = 11.9

• Q9: ∠G and ∠H are adjacent. The sum of their measures is 150°. ∠G measures (5x − 10) and ∠H is three times the size of ∠G. Find x.

• Solution: (5x − 10) + 3(5x − 10) = 150 ; (5x − 10) + (15x − 30) = 150 ; 20x − 40 = 150; 20x = 190; x = 9.5

• Q10: ∠M and ∠N are adjacent. Their sum is 135°. ∠M = (4x+8)° and ∠N is half the size of ∠M. Find x.

• Solution: (4x + 8) + 1/2(4x + 8) = 135 ; (4x + 8) + (2x + 4) = 135 ; 6x + 12 = 135 ; 6x = 123 ; x = 20.5

This quiz focuses on solving two-step equations involving angle measures, including supplementary, complementary, and adjacent angles. Each question will challenge you to apply algebraic techniques to find the unknown variable in angle relationships. Get ready to sharpen your skills in both geometry and algebra!