Question

(b) in the figure below, ( mangle xzy = 51^circ ) and ( moverarc{xy} = 37^circ ). find ( moverarc{vw} ).

Explanation:

Step1: Recall the theorem for angles formed by two secants.

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 \( m\angle XZY=\frac{1}{2}(m\widehat{VW}-m\widehat{XY}) \).

Step2: Substitute the known values into the formula.

We know that \( m\angle XZY = 51^\circ \) and \( m\widehat{XY}=37^\circ \). Substituting these into the formula: \( 51=\frac{1}{2}(m\widehat{VW}-37) \).

Step3: Solve for \( m\widehat{VW} \).

First, multiply both sides of the equation by 2: \( 51\times2=m\widehat{VW}-37 \), which simplifies to \( 102 = m\widehat{VW}-37 \). Then, add 37 to both sides: \( m\widehat{VW}=102 + 37 \).

Step4: Calculate the result.

\( 102+37 = 139 \), so \( m\widehat{VW}=139^\circ \).

Answer:

\( 139 \)