Shape and Space 2

4 marks · Hard

In the diagram below, ABCD is a cyclic quadrilateral inside a circle with centre O.

Cyclic quadrilateral with angles marked.

Angle BAD = 82° and angle BCD = 71°.

Calculate the size of angle ABC, giving a reason for each step of your working.

Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.

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Angle ABC = 98°, with explained reasoning (using cyclic quadrilateral angle facts). degrees (±0)

Also accept: Angle ABC = 98°, with correct cyclic quadrilateral argument; Angle ABC = 98; shown via calculation

Common mistakes: Forgetting to use sum of opposite angles is 180°.; Adding or not subtracting correctly.

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Marking points

M1 · 1 mark · States or uses the property that opposite angles in a cyclic quadrilateral add to 180°.

M2 · 1 mark · Forms correct equation: angle BAD + angle BCD = 180°

A1 · 1 mark · Correct calculation: 180° – (71°) = 109° for the pair BAD + BCD

A2 · 1 mark · Finds ABC as 180° – 82° = 98° and clearly states this as the answer with justification.

Worked solution

In a cyclic quadrilateral, opposite angles sum to 180°.

Given angle BAD = 82° and angle BCD = 71°.

So, 82° + angle BCD = 180°, so angle BCD = 180° – 82° = 98°.

But we are told angle BCD = 71°, so the other pair of opposite angles (ABC + CDA) sum to 180°.

Angle ABC = 180° – angle CDA. Since we know only BAD and BCD, by cyclic property:

So to find angle ABC: 82° + 71° = 153°, the sum of these two. The other two must sum to 360° – 153° = 207°. But the sum of all four angles is 360°, and for each opposite angle pair, sum is 180°.