Cyclic Quadrilaterals Examples

What measure of ∠2 is required to make quadrilateral ABCD cyclic?

We know that opposite angles of cyclic quadrilaterals must be supplementary. Since ∠2 and the given angle of 70° are opposite, they must be supplementary in order for the quadrilateral to be cyclic. Supplementary angles add up to 180°, so we can find the measure of ∠2 by setting up this equation:

m∠2 + 70° = 180°
m∠2 = 180° – 70°
m∠2 = 110°

Are all rhombi cyclic?

Let's reword the question to this: will all of a rhombus's vertices always lie perfectly on a circle? If you aren't sure, we can ask the question a third way: will opposite angles of a rhombus always be supplementary?

Rhombi have four congruent sides, parallel pairs of sides, and congruent opposite angles. The opposite angles of rhombus ABCD are only supplementary if it's really a square. (In order for two angles to be both congruent and supplementary, they both have to be 90°.)

But ABCD is still a rhombus and its opposite angles are definitely not supplementary. So the answer is no, not all rhombi are cyclic.

Which of these quadrilaterals are cyclic?

A rectangle is always cyclic. The second and third quadrilaterals here are both kites, but only the middle guy has a pair of opposite angles that are supplementary. Because of that, the first and second shapes are cyclic quadrilaterals, but we don't have enough info to tell whether the third one is.

Quadrilateral ABCD is cyclic, and ∠ABC is supplementary to ∠CBE. What is the measure of ∠CBE?

We know that opposite angles of cyclic quadrilaterals are supplementary. That's old news, but it means that the measure of ∠ABC is 80°. Since ∠ABC and ∠CBE are supplementary as well, we know that m∠CBE = 100°. Easy peasy, right?