Question

which equation is correct regarding the measure of $\angle 1$?
$\bigcirc \\ m\angle 1 = \frac{1}{2}(a - c)$
$\bigcirc \\ m\angle 1 = \frac{1}{2}(a + c)$
$\bigcirc \\ m\angle 1 = \frac{1}{2}(b - d)$
$\bigcirc \\ m\angle 1 = \frac{1}{2}(b + d)$

Explanation:

Step1: Recall the theorem for angle formed by a tangent and a secant

The measure of an angle formed by a tangent and a secant outside a circle is half the difference of the measures of the intercepted arcs. The larger arc is \( b \) and the smaller intercepted arc is \( d \).

Step2: Apply the theorem

Using the theorem, the measure of \( \angle 1 \) is \( \frac{1}{2}(b - d) \).

Answer:

\( m\angle1=\frac{1}{2}(b - d) \) (the option with \( m\angle1=\frac{1}{2}(b - d) \))