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taurus [48]
3 years ago
11

Given: ABCD is a trapezoid, AB=CD, MN is a midsegment, MN=30, BC=17, AB=26 Find: m∠A, m∠B, m∠C, and m∠D

Mathematics
1 answer:
enyata [817]3 years ago
3 0

Answer:

The measures of angle ∠A, ∠B, ∠C, and ∠D are 60, 120, 120 and 60 degree respectively.

Step-by-step explanation:

Given information:AB=CD, MN is a midsegment, MN=30, BC=17, AB=26

Since two opposites sides are equal, therefore we can say that two no parallel sides are equal.

The length of midsegment is average of length of parallel lines.

MN=\frac{AD+BC}{2}

30=\frac{AD+17}{2}

60=AD+17

43=AD

Draw perpendiculars on AD from B and C. Let angle A be θ. D

AD=AE+EF+FD

Since ABCD is an isosceles trapezoid, therefore AE=FD and EF=BC

AD=AE+EF+AE

43=2(AE)+17

26=2(AE)

13=AE

\cos\theta=\frac{base}{hypotenuse}

\cos\theta=\frac{AE}{AB}

\cos\theta=\frac{13}{26}

\cos\theta=\frac{1}{2}

\theta=\cos^{-1}\frac{1}{2}

\theta=60

Since ABCD is an isosceles trapezoid, therefore angles A and D are same. Angle B and C are same.

\angle A=\angle D=60^{\circ}

The sum of two consecutive angles of a trapezoid is 180 degree by consecutive interior angle theorem.

\angle A+\angle B=180^{\circ}

60^{\circ}+\angle B=180^{\circ}

\angle B=120^{\circ}

\angle B=\angle C=120^{\circ}

Therefore measures of angle ∠A, ∠B, ∠C, and ∠D are 60, 120, 120 and 60 degree respectively.

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