Question

Draw the following diagram: quadrilateral ABCD
inscribed in circle E
. The mAB=54°
, mBC=115°
, and mCD=86°
. Select the true statement.(1 point)
Responses

mADC=84.5°

m Modifying above upper A upper D upper C with arc equals 84.5 degrees

m∠B=95.5°

m angle upper B equals 95.5 degrees

m∠D=169°

m angle upper D equals 169 degrees

mAD=115°

Answer 1

Interpret the given as arc measures: m(arc AB)=54°, m(arc BC)=115°, m(arc CD)=86°. Then m(arc DA)=360−(54+115+86)=105°.

Angle B (∠ABC) intercepts arc AC that does NOT contain B, i.e. arc A→D→C = arc AD+arc DC = 105+86 = 191°. So
m∠B = 1/2 · 191° = 95.5°.

Therefore the true statement is: m∠B = 95.5°.

(For reference: m∠D = 1/2·(arc AC via B) = 1/2·(54+115)=84.5°, and arc AD = 105°, so the other choices are false.)