12 multiple-choice questions, progressively harder.
In convex cyclic ABCDABCDABCD, the whole angles at A,CA,CA,C measure (4x+9)∘(4x+9)^\circ(4x+9)∘ and (2x+3)∘(2x+3)^\circ(2x+3)∘. Find xxx.
Solution
Correct answer: C
The opposite angles are supplementary.
The resulting angles are 121 and 59 degrees, both valid convex angles.
In convex cyclic ABCDABCDABCD, angles ABC,CDAABC,CDAABC,CDA measure (5x−8)∘(5x-8)^\circ(5x−8)∘ and (3x+12)∘(3x+12)^\circ(3x+12)∘. Find xxx.
Correct answer: A
B and D are opposite.
The angles are 102 and 78 degrees, so their measures are valid.
In convex cyclic ABCDABCDABCD, side BCBCBC is extended through CCC to EEE. Exterior angle DCEDCEDCE measures (3x+5)∘(3x+5)^\circ(3x+5)∘, and interior angle BADBADBAD measures (5x−23)∘(5x-23)^\circ(5x−23)∘. Find xxx.
Correct answer: D
The exterior angle equals the opposite interior angle.
Both named angles then measure 47 degrees.
Convex cyclic ABCDABCDABCD has m∠A=82∘m\angle A=82^\circm∠A=82∘ and m∠B=119∘m\angle B=119^\circm∠B=119∘. Which pair gives the whole angles at C,DC,DC,D?
m∠C=98∘m\angle C=98^\circm∠C=98∘
m∠D=119∘m\angle D=119^\circm∠D=119∘
m∠D=61∘m\angle D=61^\circm∠D=61∘
m∠C=82∘m\angle C=82^\circm∠C=82∘
Correct answer: B
Subtract each given angle from 180 degrees using the opposite pair.
Convex ABCDABCDABCD has diagonal ACACAC. The givens are m∠BAC=31∘m\angle BAC=31^\circm∠BAC=31∘, m∠BCA=65∘m\angle BCA=65^\circm∠BCA=65∘, and m∠CAD=42∘m\angle CAD=42^\circm∠CAD=42∘. What value of m∠ACDm\angle ACDm∠ACD makes the quadrilateral cyclic?
Add the two pieces at A to obtain 73 degrees. The whole angle at C must therefore be 107 degrees. Its known piece is 65 degrees.
With this value the opposite-angle sum is 180 degrees, so the converse applies.
Convex cyclic ABCDABCDABCD has m∠BCD=118∘m\angle BCD=118^\circm∠BCD=118∘. Diagonal AC bisects the whole angle BAD. Find m∠CADm\angle CADm∠CAD.
The whole angle at A is supplementary to the whole angle at C. Diagonal AC then splits the whole angle at A into two equal parts.
Convex ABCDABCDABCD has BCBCBC extended through CCC to EEE. Both ∠DCE\angle DCE∠DCE and ∠BAD\angle BAD∠BAD measure 68∘68^\circ68∘. Which conclusion follows?
The linear pair makes the whole angle at C 112 degrees. It is supplementary to the opposite angle at A.
The converse proves cyclicity.
Only consecutive angles A=77∘A=77^\circA=77∘ and B=103∘B=103^\circB=103∘ are known in convex ABCDABCDABCD. What can be decided about cyclicity from these data alone?
The converse needs an opposite pair. A and B are consecutive, so their supplementary sum does not settle cyclicity. For instance, the remaining angles could be C=103, D=77, satisfying the cyclic test, or C=77, D=103 in an oblique parallelogram, failing it.
In convex cyclic ABCDABCDABCD, the whole angle at A is twice the whole angle at C. Find the whole angle at C.
Write the angle at C as t degrees. The opposite angle at A is 2t degrees.
Convex cyclic ABCDABCDABCD has AB=ADAB=ADAB=AD and m∠BCD=112∘m\angle BCD=112^\circm∠BCD=112∘. Find m∠ABDm\angle ABDm∠ABD.
The opposite whole angle BAD is 68 degrees. In isosceles triangle ABD, the base angles at B,D are equal.
Convex cyclic ABCDABCDABCD has AB∥CDAB\parallel CDAB∥CD and m∠DAB=64∘m\angle DAB=64^\circm∠DAB=64∘. Find m∠ABCm\angle ABCm∠ABC.
Parallel bases make the same-side angles A,D supplementary along AD. Thus D is 116 degrees. The cyclic opposite pair B,D is also supplementary.
Convex cyclic ABCDABCDABCD has m∠BAD=73∘m\angle BAD=73^\circm∠BAD=73∘. Find the measure of arc DABDABDAB, the arc from D to B through A.
Angle BAD intercepts the other arc, BCD, whose measure is 146 degrees. Subtract that from the full circle.
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