Question

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

(2 points)
m∠A=
𝑚
∠
𝐴
=
°, m∠C=
𝑚
∠
𝐶
=
°, and m∠D=
𝑚
∠
𝐷
=
°

Answer 1

m∠A = 82°, m∠C = 98°, m∠D = 87°.

Work (using inscribed-angle/arc facts):
- Let arcs AB, BC, CD, DA be in order. Given arc BC = 58°, arc CD = 106°, and m∠B = 93°.
- An inscribed angle equals half its intercepted arc. ∠B intercepts arc CD + DA, so CD + DA = 2·93° = 186°. Hence DA = 186° − 106° = 80°.
- The whole circle is 360°, so AB = 360° − (BC + CD + DA) = 360° − (58° + 106° + 80°) = 116°.
- ∠A intercepts arc BD = BC + CD = 58° + 106° = 164°, so m∠A = 1/2·164° = 82°.
- ∠C intercepts arc DA + AB = 80° + 116° = 196°, so m∠C = 1/2·196° = 98°.
- ∠D intercepts arc AB + BC = 116° + 58° = 174°, so m∠D = 1/2·174° = 87°.