Question

Use the image to answer the question.

Quadrilateral upper A upper B upper C upper D is drawn within a circle such that all vertices touch the edge of the circle.

The m∠B
is 93°, mBC
is 58°, and mCD
is 106°. Find the missing angle measures.

(2 points)
m∠A=
°, m∠C=
°, and m∠D=
°

Answer 1

m∠A = 82°, m∠C = 98°, m∠D = 87°.

Work (brief):
- Inscribed angle B = 93° ⇒ intercepted arc AC = 2·93 = 186°.
- AC = AD + DC, so AD = 186 − DC = 186 − 106 = 80°.
- Total circle 360° ⇒ AB = 360 − (BC + CD + DA) = 360 − (58 + 106 + 80) = 116°.
- m∠A = 1/2 · arc BD = 1/2 · (BC + CD) = 1/2 · (58 + 106) = 82°.
- m∠C = 1/2 · arc AB + AD = 1/2 · (116 + 80) = 98°.
- m∠D = 1/2 · arc AB + BC = 1/2 · (116 + 58) = 87°.