Exercise 53 Page 528

We want to state whether the opposite angles of a trapezoid are always, sometimes, or never supplementary. Let's begin by recalling one of the theorems about isosceles trapezoids.

Let's take a look at the example isosceles trapezoid ABCD and mark its congruent angles.

From the definition of a trapezoid, we know that the bases are parallel. This means that the angles adjacent to the same leg of the trapezoid are supplementary. Knowing that in an isosceles trapezoid each pair of base angles is congruent, we can see that opposite angles are also supplementary. m ∠ DAB + m∠ BCD = 180^(∘) A trapezoid with supplementary opposite angles exists. However, the base angles of a trapezoid are not congruent in the case of a non-isosceles trapezoid. This means that not all trapezoids have supplementary opposite angles. Therefore, the statement is sometimes true.