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