Question

in the diagram of circle c, what is the measure of ∠1?
17°
35°
70°
71°

Explanation:

Step1: Recall secant angle theorem

The measure of an angle formed by two secants intersecting outside a circle is half the difference of the measures of the intercepted arcs. The formula is:
$$m\angle 1 = \frac{1}{2} \times (\text{measure of larger arc} - \text{measure of smaller arc})$$

Step2: Identify intercepted arcs

The larger intercepted arc is $106^\circ$, the smaller intercepted arc is calculated as:
$$180^\circ - 106^\circ - 36^\circ = 38^\circ$$

Step3: Calculate $\angle 1$

Substitute the arc measures into the formula:
$$m\angle 1 = \frac{1}{2} \times (106^\circ - 38^\circ) = \frac{1}{2} \times 68^\circ = 34^\circ$$
*Note: Correcting for standard circle arc logic: the smaller intercepted arc corresponding to the $36^\circ$ inscribed angle is $2\times36^\circ=72^\circ$. Recalculating:
$$m\angle 1 = \frac{1}{2} \times (106^\circ - 72^\circ) = \frac{1}{2} \times 34^\circ = 17^\circ$$*

Answer:

17°