Question

question 33 of 44 which of the following polygons can be circumscribed by a circle? polygon a (vertices d, e, f, g) has angles: ∠d=120°, ∠e=60°, ∠f=120°, ∠g=60°; polygon b (vertices a, b, c, d) has angles: ∠b=90°, ∠c=90°, ∠d=90°, ∠a=90°. options: a. polygon a because its adjacent angles are different but supplementary. b. both polygon a and polygon b since both are parallelograms. c. neither polygon because circles cannot be circumscribed about quadrilaterals. d. polygon b because its opposite angles are supplementary.

Explanation:

Step1: Recall cyclic quadrilateral condition

A quadrilateral can be circumscribed by a circle (cyclic quadrilateral) if and only if the sum of each pair of opposite angles is \(180^\circ\) (supplementary).

Step2: Analyze Polygon A (parallelogram)

In Polygon A, angles are \(120^\circ, 60^\circ, 120^\circ, 60^\circ\). Opposite angles: \(120^\circ\) and \(120^\circ\) (sum \(240^\circ\)), \(60^\circ\) and \(60^\circ\) (sum \(120^\circ\)). Not supplementary. So not cyclic.

Step3: Analyze Polygon B (rectangle, a type of parallelogram)

In Polygon B (rectangle), all angles are \(90^\circ\). Opposite angles: \(90^\circ + 90^\circ = 180^\circ\) (supplementary). So it is cyclic.

Step4: Evaluate options

• Option A: Polygon A is not cyclic (opposite angles not supplementary), so A is wrong.
• Option B: Polygon A is not cyclic, so B is wrong.
• Option C: Circles can be circumscribed about some quadrilaterals (e.g., rectangles), so C is wrong.
• Option D: Polygon B (rectangle) has opposite angles supplementary (\(90^\circ+90^\circ = 180^\circ\)), so it can be circumscribed by a circle. This is correct.

Answer:

D. Polygon B because its opposite angles are supplementary.