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

Compute the missing arc AD first:
m(arc AD) = 360° − (54° + 115° + 86°) = 360° − 255° = 105°.

Now test each statement using the Inscribed Angle Theorem (an inscribed angle = 1/2 the measure of its intercepted arc).

1) m(arc AD) = 115° — false (we found 105°).

2) m∠B = 95.5° — true.
∠B intercepts arc ADC = arc AD + arc DC = 105° + 86° = 191°, so m∠B = 1/2·191° = 95.5°.

3) m∠D = 169° — false.
∠D intercepts arc AC = arc AB + arc BC = 54° + 115° = 169°, so m∠D = 1/2·169° = 84.5°, not 169°.

4) m(arc ADC) = 84.5° — false.
Arc ADC (the arc from A through D to C) is 105° + 86° = 191°, not 84.5°. (84.5° is m∠D.)

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