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.
[tex]MN=\frac{AD+BC}{2}[/tex]
[tex]30=\frac{AD+17}{2}[/tex]
[tex]60=AD+17[/tex]
[tex]43=AD[/tex]
Draw perpendiculars on AD from B and C. Let angle A be θ. D
[tex]AD=AE+EF+FD[/tex]
Since ABCD is an isosceles trapezoid, therefore AE=FD and EF=BC
[tex]AD=AE+EF+AE[/tex]
[tex]43=2(AE)+17[/tex]
[tex]26=2(AE)[/tex]
[tex]13=AE[/tex]
[tex]\cos\theta=\frac{base}{hypotenuse}[/tex]
[tex]\cos\theta=\frac{AE}{AB}[/tex]
[tex]\cos\theta=\frac{13}{26}[/tex]
[tex]\cos\theta=\frac{1}{2}[/tex]
[tex]\theta=\cos^{-1}\frac{1}{2}[/tex]
[tex]\theta=60[/tex]
Since ABCD is an isosceles trapezoid, therefore angles A and D are same. Angle B and C are same.
[tex]\angle A=\angle D=60^{\circ}[/tex]
The sum of two consecutive angles of a trapezoid is 180 degree by consecutive interior angle theorem.
[tex]\angle A+\angle B=180^{\circ}[/tex]
[tex]60^{\circ}+\angle B=180^{\circ}[/tex]
[tex]\angle B=120^{\circ}[/tex]
[tex]\angle B=\angle C=120^{\circ}[/tex]
Therefore measures of angle ∠A, ∠B, ∠C, and ∠D are 60, 120, 120 and 60 degree respectively.
