Question

what is the measure of $overline{au}$?○ $44^circ$○ $50^circ$○ $64^circ$○ $92^circLXB0111^circ$

Explanation:

Step1: Recall inscribed angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc. So, $\angle QDU = \frac{1}{2}m\overset{\frown}{QU}$

Step2: Calculate $\angle QDU$

Substitute $m\overset{\frown}{QU}=88^\circ$:
$\angle QDU = \frac{1}{2} \times 88^\circ = 44^\circ$

Step3: Use cyclic quadrilateral property

Opposite angles in a cyclic quadrilateral sum to $180^\circ$. So $\angle QDU + \angle UAQ = 180^\circ$

Step4: Calculate $\angle UAQ$

$\angle UAQ = 180^\circ - 44^\circ = 136^\circ$

Step5: Find arc sum for quadrilateral

The total sum of arcs in a circle is $360^\circ$. Let $m\overset{\frown}{AU}=x$, $m\overset{\frown}{QD}=y$. For cyclic quadrilateral, $\angle U = \frac{1}{2}m\overset{\frown}{QDA}$, so $111^\circ = \frac{1}{2}(88^\circ + y)$

Step6: Solve for $m\overset{\frown}{QD}$

$y = 2\times111^\circ - 88^\circ = 222^\circ - 88^\circ = 134^\circ$

Step7: Solve for $m\overset{\frown}{AU}$

$88^\circ + x + m\overset{\frown}{DA} + 134^\circ = 360^\circ$. Also, $\angle UAQ = \frac{1}{2}(m\overset{\frown}{QD} + m\overset{\frown}{DA})$, so $136^\circ = \frac{1}{2}(134^\circ + m\overset{\frown}{DA})$, $m\overset{\frown}{DA}=2\times136^\circ -134^\circ=138^\circ$. Now substitute back:
$88^\circ + x + 138^\circ + 134^\circ = 360^\circ$
$x = 360^\circ - (88^\circ+138^\circ+134^\circ) = 360^\circ - 360^\circ + 92^\circ$ (simplify directly: $x=360-88-138-134=92^\circ$)

Answer:

92°