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Cyclic Quadrilaterals: Practice

12 multiple-choice questions, progressively harder.

Level 3 · Challenge 0 / 12 answered

Question 1 of 12

1

In convex cyclic $ABCD$ , whole angles A,C,B measure $(3x+7)^\circ$ , $(7x-17)^\circ$ , and $(2x+30)^\circ$ , respectively. Find the whole angle at D.

Answer choices for question 1 A $68^\circ$ B $116^\circ$ C $64^\circ$ D $112^\circ$
2

Convex cyclic $ABCD$ has $BC$ extended through C to E. Exterior angle $DCE$ measures $(4x-6)^\circ$ , opposite interior angle BAD measures $(2x+32)^\circ$ , and angle ABC measures $(3x+5)^\circ$ . Find the whole angle CDA.

Answer choices for question 2 A $62^\circ$ B $118^\circ$ C $70^\circ$ D $110^\circ$
3

In convex $ABCD$ , triangle ABC has $m\angle BAC=36^\circ$ and $m\angle ABC=81^\circ$ . Also $m\angle CAD=27^\circ$ and $m\angle ACD=54^\circ$ . Which conclusion has a complete valid reason?

Answer choices for question 3 A It is cyclic because the two pieces at A are equal. B It is not cyclic because angle ABC is not right. C It is cyclic because the whole opposite angles A,C total $180^\circ$ . D It is not cyclic because the whole opposite-angle sum is $179^\circ$ .
4

In convex $ABCD$ , triangle ABC has $m\angle BAC=24^\circ$ and $m\angle ABC=92^\circ$ . Also $m\angle CAD=39^\circ$ and $m\angle ACD=52^\circ$ . Is the quadrilateral cyclic, and what is the whole opposite-angle sum A+C?

Answer choices for question 4 A No; $179^\circ$ . B Yes; $179^\circ$ . C No; $180^\circ$ . D Yes; $180^\circ$ .
5

Convex cyclic $ABCD$ has $m\angle BAC=29^\circ$ , $m\angle CAD=46^\circ$ , and $m\angle BCA=61^\circ$ . Find $m\angle ABD$ .

Answer choices for question 5 A $46^\circ$ B $61^\circ$ C $44^\circ$ D $105^\circ$
6

Convex cyclic $ABCD$ has $m\angle BAD=100^\circ$ , $m\angle ABD=29^\circ$ , and $m\angle DBC=33^\circ$ . Find $m\angle BDC$ .

Answer choices for question 6 A $51^\circ$ B $80^\circ$ C $118^\circ$ D $67^\circ$
7

Noncollinear points A,B,C determine a circle. A point D makes convex $ABCD$ , and $m\angle ABC=115^\circ$ , $m\angle CDA=65^\circ$ . Must D lie on the circle through A,B,C?

Answer choices for question 7 A No, because an acute angle at D cannot occur in a cyclic quadrilateral. B Yes, by the opposite-angle converse and uniqueness of the circle through A,B,C. C Yes, because any fourth point in a convex quadrilateral lies on the first three points' circle. D No, because only opposite right angles can prove cyclicity.
8

In convex cyclic $ABCD$ , whole angles A,C measure $3t$ degrees and $2t$ degrees. Whole angle D measures $(2t+5)^\circ$ . Extend AB through B to E. Find exterior angle CBE.

Answer choices for question 8 A $77^\circ$ B $103^\circ$ C $72^\circ$ D $108^\circ$
9

Convex cyclic $ABCD$ has equal whole angles at A,C. Diagonal BD splits angle ABC into $35^\circ$ and $46^\circ$ . Which segment must be a diameter of its circle?

Answer choices for question 9 A $AC$ B $AB$ C $CD$ D $BD$
10

A convex isosceles trapezoid ABCD has chosen parallel bases AB and CD. Which reasoning proves that it is cyclic?

Answer choices for question 10 A The bases are parallel, so A and C must be equal; equal opposite angles always prove cyclicity. B Its legs are equal, so every vertex angle is right. C Base angles A,B are equal, and $B+C=180^\circ$ , so $A+C=180^\circ$ ; apply the cyclicity converse. D Every convex quadrilateral is cyclic because its angles total $360^\circ$ .
11

Convex cyclic $ABCD$ has $AB=BC$ and $m\angle BAC=27^\circ$ . Find the whole interior angle CDA.

Answer choices for question 11 A $54^\circ$ B $126^\circ$ C $27^\circ$ D $153^\circ$
12

In the converse proof, X lies strictly inside chord AC of the circle through A,B,C. Line BX has another circle intersection E. Which point order follows, and why?

Answer choices for question 12 A B, E, X, because a second intersection must be nearer to B than every interior point. B B, X, E, because X is strictly inside the circle and lies between its two intersections with the line. C X, B, E, because a circle point lies between an interior point and the other circle point. D No second intersection is guaranteed, because the line could be tangent while containing X.

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