Quadrilaterals Exercises

Are all rectangles cyclic quadrilaterals? Why or why not?

Hint

What do we know about the internal angles of rectangles?

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?

When, if ever, is a parallelogram cyclic?

Hint

What do we know about the opposite angles of a parallelogram?

Show that an isosceles trapezoid is always cyclic.

Hint

Show that the sum of a pair of opposite angles equals 180 degrees.

All cyclic quadrilaterals have diagonals that are congruent. Is this true or false?

Hint

Can you think of or draw a counterexample?

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.

Statements	Reasons
1. THEO is a cyclic quadrilateral	Given
2. mTHE + mTOE = 360°	Arc addition postulate (1)
3. m∠THE = ½ × mTHE and m∠TOE = ½ × mTOE	Inscribed Angle Theorem (1)
4. 2(m∠THE) = mTHE and 2(m∠TOE) = mTOE	Multiplication Property of Equality (3)
5. 2(m∠THE) + 2(m∠TOE) = 360°	Substitution (2, 4)
6. m∠THE + m∠TOE = 180°	Division Property of Equality (5)
7. ∠THE and ∠TOE are supplementary	Definition of supplementary angles (6)