Exercise 30 Page 559

We want to determine whether an isosceles trapezoid can always be inscribed inside a circle. To do it, let's recall the Inscribed Quadrilateral Theorem, Theorem 10.13.

Inscribed Quadrilateral Theorem

A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

To consider the relationship between angles in an isosceles trapezoid, let's recall the Isosceles Trapezoid Base Angles Theorem, Theorem 7.14.

Isosceles Trapezoid Base Angles Theorem

If a trapezoid is isosceles, then each pair of base angles is congruent.

Let's look at an example of an isosceles trapezoid.

By the Isosceles Trapezoid Base Angles Theorem we know that m∠ A = m∠ D and m∠ B = m∠ C. Since sum of the interior angles in every quadrilateral is 360 ^(∘), we can write the following equation. m∠ A + m∠ B + m∠ C + m∠ D =360 ^(∘) As we can see, ∠ A and ∠ C are opposite angles. We will calculate the sum of their measures using the equation above. The above equation means that ∠ A and ∠ C are supplementary. Notice that the same reasoning works for the second pair of opposite angles — ∠ B and ∠ D. The opposite angles are supplementary angles. By the Inscribed Quadrilateral Theorem, an isosceles trapezoid can always be inscribed in a circle.