Question

what is the value of z?
○ 52°
○ 104°
○ 116°
○ 208°
120°
112°

Explanation:

Step1: Recall circle angle sum rule

The total degrees in a circle is $360^\circ$.

Step2: Sum known arc angles

Add the given arc measures: $120^\circ + 112^\circ = 232^\circ$

Step3: Find remaining arc pair sum

Subtract from total circle degrees: $360^\circ - 232^\circ = 128^\circ$

Step4: Use vertical angle arc property

Arc $z$ and arc $AED$ are vertical angle arcs? No, correct: Arc $z$ corresponds to the vertical angle's arc? Wait, no: The inscribed angle $\angle AEB = 112^\circ$, so its intercepted arc is $120^\circ + z$? No, correct theorem: The measure of an angle formed by two intersecting chords is half the sum of the measures of the intercepted arcs.
$\angle AEB = \frac{1}{2}(\text{arc } AB + \text{arc } CD)$
$112^\circ = \frac{1}{2}(120^\circ + z)$
Multiply both sides by 2: $224^\circ = 120^\circ + z$
Solve for $z$: $z = 224^\circ - 120^\circ = 104^\circ$

Answer:

104°