Given:
Measure of arc UV is 56 degrees.
The measure of arc QW is 132 degrees.
To find:
The measure of angle VPW.
Solution:
Intersecting chords theorem: If two chords intersect each other inside the circle, then the measure of angle on the intersection is half of the sum of intercepted arcs.
Using the intersecting chords theorem, we get
[tex]m\angle VPU=\dfrac{1}{2}(m(arcQW)+m(arcUV))[/tex]
Substituting the given values, we get
[tex]m\angle VPU=\dfrac{1}{2}(132^\circ+56^\circ)[/tex]
[tex]m\angle VPU=\dfrac{1}{2}(188^\circ)[/tex]
[tex]m\angle VPU=94^\circ[/tex]
Now,
[tex]m\angle VPU+m\angle VPW=180^\circ[/tex] [Linear pair]
[tex]94^\circ+m\angle VPW=180^\circ[/tex]
[tex]m\angle VPW=180^\circ-94^\circ[/tex]
[tex]m\angle VPW=86^\circ[/tex]
Therefore, the measure of angle VPW is [tex]86^\circ[/tex].
