Question

find $m\angle1$.
$63^\circ$
$126^\circ$
$54^\circ$
$27^\circ$

Explanation:

Step1: Recall external angle theorem

The measure of an angle formed outside a circle by two secants is half the difference of the measures of the intercepted arcs. The intercepted arcs are the major arc (126°) and its corresponding minor arc. First, calculate the minor arc:
$$360^\circ - 126^\circ = 234^\circ$$

Step2: Calculate $m\angle1$

Apply the external angle formula:
$$m\angle1 = \frac{1}{2} \times (234^\circ - 126^\circ)$$
$$m\angle1 = \frac{1}{2} \times 108^\circ$$

Answer:

$54^\circ$ (Green option: $54^\circ$)