Question

(a) in the figure below, ( moverarc{vw} = 81^circ ) and ( moverarc{xy} = 115^circ ). find ( mangle vzw ).
( mangle vzw = square^circ )

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

Explanation:

Step1: Recall the theorem for angle 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 \( m\angle XZY=\frac{1}{2}(m\overarc{VW}-m\overarc{XY}) \).

Step2: Substitute the known values

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

Step3: Solve for \( m\overarc{VW} \)

Multiply both sides by 2: \( 51\times2=m\overarc{VW}-37 \)
\( 102 = m\overarc{VW}-37 \).
Add 37 to both sides: \( m\overarc{VW}=102 + 37 \)
\( m\overarc{VW}=139 \).

Answer:

\( 139 \)