In a circle, chords WY and XZ intersect at point P. The measure of minor arc WX is 55^\cir… — SAT

In a circle, chords $WY$ and $XZ$ intersect at point $P$ . The measure of minor arc $WX$ is $55^\circ$ and the measure of minor arc $YZ$ is $105^\circ$ . What is the measure, in degrees, of angle $WPX$ ?

Answer: 80

Answer

The correct answer is 80. According to the intersecting chords angle theorem, when two chords intersect inside a circle, the measure of the angle they form is half the sum of the measures of the intercepted arcs. Here, angle

WPX

and its vertical angle intercept minor arcs

WX

and

YZ

. Therefore, the measure of angle

WPX

\frac{55^\circ + 105^\circ}{2} = \frac{160^\circ}{2} = 80^\circ

Step-by-Step Solution

Identify the geometric relationship for angles formed by intersecting chords inside a circle.

The measure of

\angle WPX

is equal to half the sum of the measures of its intercepted arc

WX

and the intercepted arc of its vertical angle, arc

YZ

By the intersecting chords angle theorem, the angle formed by two intersecting chords inside a circle is half the sum of the measures of the intercepted arcs.

Sum the measures of the intercepted arcs.

55^\circ + 105^\circ = 160^\circ

The measures of minor arcs

WX

and

YZ

are given as

55^\circ

and

105^\circ

respectively.

Divide the sum of the arc measures by 2.

80

Halving the sum of the arc measures (

160^\circ

) yields the measure of the angle:

\frac{160^\circ}{2} = 80^\circ

Key Concept

Intersecting Chords Angle Theorem

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