Question

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

Explanation:

Step1: Recall the formula for the measure of an angle formed by two secants

The measure of an angle formed by two secants outside a circle is half the difference of the measures of the intercepted arcs.

Step2: Identify the intercepted arcs

In the given figure, the angle \(∠1\) is formed by two secants. The intercepted arcs are arc \(b\) and arc \(d\).

Step3: Apply the formula

By the formula \(m\angle=\frac{1}{2}(\text{larger arc}-\text{smaller arc})\), for \(∠1\), we have \(m\angle1 = \frac{1}{2}(b - d)\)

Answer:

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