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  1. Problems
  2. /The angles A, B and C of a cyclic quadrilateral ABCD are in the ratio 1 : 2 : 5…
MediumMathematicsHigh School (9–10)GeneralGeometry › CirclesNumerical

The angles A, B and C of a cyclic quadrilateral ABCD are in the ratio 1 : 2 : 5…

StemPy LabContributed by StemPy Lab

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

The angles AAA, BBB and CCC of a cyclic quadrilateral ABCDABCDABCD are in the ratio 1:2:51 : 2 : 51:2:5. Find all four angles.

Check your answer

Work it out on paper, then type your number. The answer itself stays hidden.

Need a reminder? The key ideaShowHide

The opposite angles of a cyclic quadrilateral add up to 180∘180^\circ180∘: ∠A+∠C=180∘\angle A + \angle C = 180^\circ∠A+∠C=180∘ and ∠B+∠D=180∘\angle B + \angle D = 180^\circ∠B+∠D=180∘. The four angles of any quadrilateral add up to 360∘360^\circ360∘.

The worked solution

Have you tried it?

Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

Step 1. One part. ∠A+∠C=180∘\angle A + \angle C = 180^\circ∠A+∠C=180∘ is 666 parts, so one part is 30∘30^\circ30∘.

Step 2. Three angles. ∠A=30∘\angle A = 30^\circ∠A=30∘, ∠B=60∘\angle B = 60^\circ∠B=60∘, ∠C=150∘\angle C = 150^\circ∠C=150∘.

Step 3. The fourth angle. ∠D=180∘−∠B=120∘\angle D = 180^\circ - \angle B = 120^\circ∠D=180∘−∠B=120∘.

Step 4. Check. 30+60+150+120=36030 + 60 + 150 + 120 = 36030+60+150+120=360.

Answer: 30∘30^\circ30∘, 60∘60^\circ60∘, 150∘150^\circ150∘ and 120∘120^\circ120∘.

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← ABCD is a cyclic quadrilateral with ∠ A = 82° and ∠ B = 97°. Find ∠ C and ∠ D.The opposite angles of a cyclic quadrilateral are (3x + 10)° and (5x - 6)°… →
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  1. Cyclic quadrilaterals and their anglesCyclic Quadrilaterals

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