Question

Draw the following diagram: quadrilateral $ABCD$ inscribed in circle $E$. The $m\overparen{AB} = 54^\circ$, $m\overparen{BC} = 115^\circ$, and $m\overparen{CD} = 86^\circ$. Select the true statement.$m\overparen{AD} = 115^\circ$$m\angle B = 95.5^\circ$ $m\angle D = 169^\circ$$m\overparen{ADC} = 84.5^\circ$

Answer 1

First find the missing arc AD:
m(arc AB)+m(arc BC)+m(arc CD)+m(arc AD)=360°
54°+115°+86°+m(arc AD)=360° ⇒ m(arc AD)=360−255=105°.

Now compute inscribed angles (an inscribed angle = half the measure of its intercepted arc).

- ∠B intercepts arc A–C that goes through D: m(arc ADC)=m(arc AD)+m(arc DC)=105°+86°=191°, so
m∠B = 1/2·191° = 95.5°.

- ∠D intercepts arc C–A that goes through B: m(arc CBA)=115°+54°=169°, so
m∠D = 1/2·169° = 84.5°.

Check the choices:
- m(arc AD) = 115° — false (it is 105°).
- m∠B = 95.5° — true.
- m∠D = 169° — false (that 169° is the intercepted arc; ∠D = 84.5°).
- m(arc ADC) = 84.5° — false (arc ADC = 191°; 84.5° equals ∠ADC).

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