Applying the properties of a trapezoid and isosceles trapezoid, the missing measures are:
1. m∠C = 101°
m∠E = 46°
2. m∠Q = 89°
m∠S = 153°
3. m∠J = 83°
m∠L = 97°
m∠M = 97°
4. m∠W = 34°
m∠X = 34°
m∠Z = 146°
5. x = 4
6. m∠B = 119°
7. m∠M = 128°
m∠N = 128°
m∠O = 52°
m∠P = 52°
What are the Properties of a Trapezoid?
- The two top angles in an isosceles trapezoid are congruent, as well as the two bottom angles.
- In a trapezoid, angles that lie on the same leg (adjacent angles) are supplementary.
- In an isosceles trapezoid, opposite angles are supplementary.
Using the above properties of a trapezoid, we would solve the given problem as shown below:
1. m∠C = 180 - 79 = 101° (adjacent angles)
m∠E = 180 - 134 = 46° (adjacent angles)
2. m∠Q = 180 - 91 = 89° (adjacent angles)
m∠S = 180 - 27 = 153° (adjacent angles)
3. m∠J = m∠K = 83° (isosceles trapezoid)
m∠L = 180 - 83 = 97° (opposite angles in an isosceles trapezoid)
m∠M = m∠L = 97°
4. m∠W = 180 - 146 = 34° (opposite angles in an isosceles trapezoid)
m∠X = m∠W = 34°
m∠Z = m∠Y = 146°
5. 14x - 15 + 139 = 180 (supplementary angles)
14x + 124 = 180
14x = 180 - 124
14x = 56
x = 4
6. 9x + 2 + 5x - 4 = 180 (supplementary angles)
14x - 2 = 180
14x = 180 + 2
14x = 182
x = 13
m∠B = 9x + 2
Plug in the value of x
m∠B = 9(13) + 2
m∠B = 119°
7. 8x - 16 = 6x + 20 (isosceles trapezoid)
Find x
8x - 6x = 16 + 20
2x = 36
x = 18
m∠M = 8x - 16
Plug in the value of x
m∠M = 8(18) - 16 = 128°
m∠N = m∠M = 128°
m∠O = 180 - 128 = 52° (supplementary angles)
m∠P = m∠O = 52° (congruent angles)
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