Question

which equation correctly describes the relationship between the measures of the angle and arcs formed by the intersecting secants?
○ ( mangle 1 = \frac{1}{2} moverarc{rs} )
○ ( mangle 1 = moverarc{pq} - moverarc{rs} )
○ ( mangle 1 = \frac{1}{2}(moverarc{pq} + moverarc{rs}) )
○ ( mangle 1 = \frac{1}{2}(moverarc{pq} - moverarc{rs}) )

Explanation:

Step1: Recall the theorem for angle formed by two secants outside a circle

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 \( m\angle1=\frac{1}{2}(m\widehat{PQ}-m\widehat{RS}) \), where \( \widehat{PQ} \) is the larger intercepted arc and \( \widehat{RS} \) is the smaller intercepted arc.

Step2: Analyze each option

• Option 1: \( m\angle1 = \frac{1}{2}m\widehat{RS} \) - This is incorrect as it does not account for the other arc and the difference.
• Option 2: \( m\angle1 = m\widehat{PQ}-m\widehat{RS} \) - This is incorrect as it does not have the factor of \( \frac{1}{2} \).
• Option 3: \( m\angle1=\frac{1}{2}(m\widehat{PQ}+m\widehat{RS}) \) - This is the formula for an angle formed by two chords intersecting inside the circle (sum of arcs), not outside. So incorrect.
• Option 4: \( m\angle1=\frac{1}{2}(m\widehat{PQ}-m\widehat{RS}) \) - This matches the theorem for angle formed by two secants outside the circle.

Answer:

\( m\angle1=\frac{1}{2}(m\widehat{PQ}-m\widehat{RS}) \) (the fourth option)