Given an isosceles trapezoid ABCD with a large base AD. Find the angles of the trapezoid if AC

| August 18, 2021 | education

Given an isosceles trapezoid ABCD with a large base AD. Find the angles of the trapezoid if AC is the bisector of the angle BAD and AD = 2 BC.

AC, by condition, is the bisector of the angle BAD. By the property of the bisector of a trapezoid, it cuts off an isosceles triangle from the lateral side. Then AB = BC = CD.

Let the base BC = X cm, then, by condition, AD = 2 * X cm.

Then AB = BC = CD = X cm.

From the top B of the trapezoid, we lower the height BH, which, on the basis of AD, cuts off the segment AH, the length of which is equal to the half-difference of the lengths of the bases.

AH = (AD – BC) / 2 = (2 * X – X) / 2 = X / 2.

In a right-angled triangle ABH leg AH = X / 2, hypotenuse AB = X cm. Since AB is twice AH, the angle against the side AH is 30, and the angle BAH = 60. Angle ABC of the trapezoid ABC = ABH + HBC = 30 + 90 = 120.

Since in an isosceles trapezoid the angles at the bases are equal, then the angle BCD = ABC = 120, the angle CDA = BAD = 60.

Answer: The angles of the trapezoid are 60 and 120.



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