Parallelograms and Triangles in Trapezoids
In trapezoid , with parallel sides and , has diagonal that bisects only . If is a right angle, how many sides in the trapezoid have equal lengths?
The figure below is not drawn to scale.
Hint: Refer to the title of the problem.
The answer is 3.
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1 solution
This is my solution and there may be simpler ways in finding the answer.
Construct BE such that point E lies on CD and the line is congruent to AD. From what we can draw, we can say that AD = BE. Since the points of the two lines are from the bases, and the bases of the trapezoid are parallel, we can say that AD and BE are parallel.
We know that AB and CD are parallel. In this case, we can say that ∠CDB is equal to ∠ABD since they are alternate interior angles. We can solve for the measure of the angle:
Now we know the measure of the missing angle. Let's focus on △DBC. It is a special right triangle. If ∠BDC is 30°, then ∠BCD is 60°. Now let's focus on △ABD. Since its base angles are congruent, the triangle is an isosceles triangle. Since it's isosceles, AD = AB = BE.
From what we have found out, we can say that quadrilateral ABED is a parallelogram, specifically, a rhombus. If opposite angles are equal, then m∠A is equal to m∠E. Since m∠A is 120° (by 180 - 60 [since ∠A and ∠D are supplementary]), m∠E is 120°.
∠BED and ∠BEC are a linear pair. If ∠BED is 120°, ∠BEC is 60°. Since BEC is an equilateral triangle (because ∠BCE and ∠BEC is 60), BE = BC = EC.
The sides that have equal measures are A D , A B , a n d B C . This is why in the trapezoid, there are three sides with equal measures.