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

36 0^{\circ},

we can write the following equation.

m ∠ A + m ∠ B + m ∠ C + m ∠ D = 36 0^{\circ}

As we can see,

∠ A

and

∠ C

are opposite angles. We will calculate the sum of their measures using the equation above.

m ∠ A + m ∠ B + m ∠ C + m ∠ D = 36 0^{\circ}

$m ∠ D = m ∠ A$ , $m ∠ B = m ∠ C$

m ∠ A + m ∠ C + m ∠ C + m ∠ A = 36 0^{\circ}

Add terms

2 m ∠ A + 2 m ∠ C = 36 0^{\circ}

$LHS / 2 = RHS / 2$

m ∠ A + m ∠ C = 18 0^{\circ}

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.

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