Question

In the figure central angle of arc APB is 120°. And the central angle of arc MQN is 50° what is the measure of <C?

• A. 70°
• B. 25°
• C. 60°
• D. 35°

Answer 1

Answer:

D. 35°

To find the measure of $\angle C$, we can use the Secant-Secant Theorem (or the intersecting secants theorem outside a circle).

The measure of $\angle C$ is $35^\circ$.

1. Identify the formula:
   When two secant lines intersect outside a circle, the measure of the angle formed ($\angle C$) is equal to half the difference of the measures of their intercepted arcs:
   $\angle C = \frac{1}{2} (\text{Arc } APB - \text{Arc } MQN)$

2. Substitute the given values:

   • Central angle of arc $APB = 120^\circ \implies \text{Measure of Arc } APB = 120^\circ$

   • Central angle of arc $MQN = 50^\circ \implies \text{Measure of Arc } MQN = 50^\circ$

3. Calculate the angle:
   $\angle C = \frac{1}{2} (120^\circ - 50^\circ)$
   $\angle C = \frac{1}{2} (70^\circ)$
   $\angle C = 35^\circ$