Cyclic Quadrilaterals: Practice Level 1 | Geometry

Level 1 · Foundational 0 / 12 answered

Question 1 of 12

1

Which description defines the cyclic quadrilaterals studied in this lesson?

Answer choices for question 1 A Any quadrilateral entirely inside a disk. B A convex quadrilateral with four distinct vertices on one circle, named in boundary order. C Any quadrilateral with three vertices on a circle. D Four circle points joined in any order, including a crossing order.
2

In convex quadrilateral $ABCD$ named in boundary order, which whole angle is opposite $\angle DAB$ ?

Answer choices for question 2 A $\angle ABC$ B $\angle CDA$ C $\angle BAC$ D $\angle BCD$
3

Convex cyclic quadrilateral $ABCD$ has $m\angle DAB=58^\circ$ . Find $m\angle BCD$ .

Answer choices for question 3 A $122^\circ$ B $58^\circ$ C $116^\circ$ D $64^\circ$
4

Convex cyclic quadrilateral $PQRS$ has $m\angle PQR=131^\circ$ . Find $m\angle RSP$ .

Answer choices for question 4 A $131^\circ$ B $41^\circ$ C $49^\circ$ D $262^\circ$
5

In convex cyclic $ABCD$ , side $BC$ is extended through $C$ to $E$ . Which interior angle equals exterior angle $DCE$ ?

Answer choices for question 5 A $\angle ABC$ B $\angle BAD$ C $\angle BCD$ D $\angle CDA$
6

In convex cyclic $ABCD$ , $B,C,E$ are collinear in that order and $m\angle DCE=84^\circ$ . Find interior angle $BCD$ .

Answer choices for question 6 A $96^\circ$ B $84^\circ$ C $42^\circ$ D $168^\circ$
7

Which relation holds for every convex cyclic quadrilateral $ABCD$ in boundary order?

Answer choices for question 7 A $m\angle A=m\angle C$ B $m\angle A+m\angle B=180^\circ$ C $m\angle A+m\angle C=360^\circ$ D $m\angle A+m\angle C=180^\circ$
8

A convex quadrilateral $ABCD$ has four distinct vertices, no three collinear, and opposite angles $A=91^\circ$ and $C=89^\circ$ . What follows?

Answer choices for question 8 A It cannot be cyclic because the angles differ. B It is a rectangle. C It is cyclic by the converse opposite-angle test. D Its diagonals must be perpendicular.
9

A convex quadrilateral $ABCD$ has whole interior angles $A=80^\circ$ and $C=101^\circ$ . Which conclusion is justified?

Answer choices for question 9 A It is not cyclic because the opposite-angle sum is $181^\circ$ . B It is cyclic because the angles total more than $180^\circ$ . C It is cyclic because the angles total less than $360^\circ$ . D Cyclicity is guaranteed whenever both angles are less than $180^\circ$ .
10

In the opposite-angle proof for cyclic $ABCD$ , angle $DAB$ intercepts arc $BCD$ and angle $BCD$ intercepts arc $DAB$ . Why do the two angle measures sum to $180^\circ$ ?

Answer choices for question 10 A Each of the two arcs must be a semicircle. B The two arcs make $360^\circ$ , and each angle is half its intercepted arc. C The two angles intercept the same arc. D All four sides must have equal length.
11

Four distinct circle points occur around the circle as $A,B,C,D$ . Which proposed quadrilateral is outside this lesson's convex boundary-order convention?

Answer choices for question 11 A $ABCD$ B $BCDA$ C $ACBD$ D $DCBA$
12

Three noncollinear vertices $A,B,C$ determine a circle. A fourth point $D$ makes convex $ABCD$ in boundary order. Which extra fact guarantees that $D$ lies on that circle?

Answer choices for question 12 A $D$ lies somewhere inside the disk. B $AB=BC$ C $m\angle ABC<180^\circ$ D $m\angle BAD+m\angle BCD=180^\circ$