Question

in the diagram of circle c, what is the measure of $angle1$?$\bigcirc$ $17^circLXB0\bigcirc$ $70^circ$$\bigcirc$ $71^circ$(diagram note: circle c with an exterior angle $angle1$, an inscribed angle of $36^circ$, and an intercepted arc of $106^circ$)

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:
$$m\angle 1 = \frac{1}{2}(m\text{outer arc} - m\text{inner arc})$$

Step2: Identify intercepted arcs

The outer intercepted arc is $106^\circ$, and the inner intercepted arc is $36^\circ$.

Step3: Calculate $\angle 1$

Substitute the arc measures into the formula:
$$m\angle 1 = \frac{1}{2}(106^\circ - 36^\circ) = \frac{1}{2}(70^\circ) = 35^\circ$$

Answer:

35°