Trapezoids Examples

Trapezoid ABCD is isosceles. What is the measure of ∠C?

Because ABCD is isosceles, we know the lower base angles are congruent and the upper base angles are congruent. This means ∠B is also 60° because it's paired up with ∠A as the lower base angle.

Since quadrilaterals have internal angles that add up to 360°, we know m∠A + m∠B + m∠C + m∠D = 360°. Plugging in 60° for m∠A and m∠B gives us m∠C + m∠D = 240°. Since these two angles are also congruent as the upper base pair of angles, each one is equal to 120°.

Therefore, m∠C = 120°.

In a trapezoid, a pair of base angles are always congruent. Is this true or false? Why?

A trapezoid is a quadrilateral with only one pair of parallel sides. Base angles could be congruent, but they don't have to be. So the statement is false. In an isosceles trapezoid, a pair of base angles are always congruent, but no other trapezoid is required to meet this criterion.

Prove that the diagonals of a trapezoid do not bisect each other.

See that word, "not"? If you weren't thinking indirect proof, you should be now. We can prove this by contradiction. We will assume that a trapezoid has diagonals that do bisect each other and show that it leads to two contradicting statements.

Let a trapezoid have diagonals bisect each other. If that's the case, then the trapezoid is also parallelogram because any quadrilateral that has diagonals that bisect each other is a parallelogram. But a parallelogram has two pairs of opposite, parallel sides. This contradicts the definition of a trapezoid, which can have only one pair of parallel sides.

This means our assumption that the diagonals bisect each other cannot possibly be true for a trapezoid.

PQ is the median of trapezoid BCDF. Given the information in the figure, find y in terms of x.

We know the median of a trapezoid has a length that's half the length of the sum of the bases. In other words, the length of the median is . The two bases FD and BC have lengths of x – 2 and x + 2, respectively. The median has a length of y. All we need to do is plug our values into the equation and isolate for y.