Are all rectangles cyclic quadrilaterals? Why or why not?
Hint
What do we know about the internal angles of rectangles?
Answer
Yes, all rectangles are cyclic quadrilaterals because their opposite angles will always be supplementary.
Example 2
ABCD is a cyclic quadrilateral and AD ≅ BC. Write a formula for the relationship between ∠1 and ∠2.
Hint
Does it matter whether we've got a rectangle or isosceles trapezoid?
Answer
m∠1 + m∠2 = 180°
Example 3
When, if ever, is a parallelogram cyclic?
Hint
What do we know about the opposite angles of a parallelogram?
Answer
A parallelogram is cyclic only when it's a rectangle.
Example 4
Show that an isosceles trapezoid is always cyclic.
Hint
Show that the sum of a pair of opposite angles equals 180 degrees.
Answer
An isosceles trapezoid has several properties. One set of opposite sides are parallel, and the other set of opposite sides are congruent. Also, the internal angles shared by a base are congruent. Let's start with those.
Trapezoid ABCD has ∠1 ≅ ∠2 and ∠3 ≅ ∠4. Since AB || CD, we know that ∠1 and ∠4 are supplementary. But because ∠3 ≅ ∠4, we know that ∠1 and ∠3 must also be supplementary. Since we have a pair of supplementary opposite angles, we know ABCD is cyclic.
Example 5
All cyclic quadrilaterals have diagonals that are congruent. Is this true or false?
Hint
Can you think of or draw a counterexample?
Answer
False. Some kites are still cyclic even though their diagonals are not congruent.
Example 6
Prove that cyclic quadrilaterals have supplementary opposite angles. (This one ain't easy.)
Hint
You'll need the Inscribed Angle Theorem, which says that the measure of an inscribed angle is half the measure of the arc it intercepts.
