We want to determine the type of a trapezoid that can be inscribed in a circle. Let's begin by recalling the definition of a trapezoid.
A trapezoid is a quadrilateral with exactly one pair of parallel sides.
Using this definition, we will draw several diagrams for a trapezoid inscribed in a circle to make a conjecture.
To determine the type of the trapezoid, we will investigate the interior angles. Note that for each diagram, since ABCD is a trapezoid, AD∥ BC. Let's consider one of the diagrams.
The opposite angles of a quadrilateral inscribed in a circle are supplementary.
Using this corollary, we can write another pair of equations.
m∠ A + m∠ C=180
m∠ B + m∠ D=180
If we combine the theorem and the corollary, we can see that each pair of base angles of the trapezoid are congruent to each other.
c|c
m∠ A+m∠ B=180 m∠ A+m∠ C=180 & m∠ A+m∠ B=180 m∠ B + m∠ D=180
⇓ & ⇓
∠ B ≅ ∠ C & ∠ A ≅ ∠ D
Therefore, by the definition of an isosceles trapezoid, the type of a trapezoid inscribed in a circle is an isosceles trapezoid.
