Question

type the correct answer in each box. use numerals instead of words. if necessary, use / for the fraction bar.
an inscribed square is constructed in circle m. what are the measures of the angles and arcs?
$moverarc{xy}=square^{circ}$
$moverarc{wy}=square^{circ}$
$mangle zwx=square^{circ}$

Explanation:

Step1: Properties of inscribed square

A square has all sides equal and all angles equal to \(90^{\circ}\). In a circle, the measure of an inscribed angle is half the measure of its intercepted arc. Also, the sum of arcs in a circle is \(360^{\circ}\). Since it's a square inscribed in a circle, the arcs between consecutive vertices are equal.

Step2: Measure of arc \(XY\)

The total degrees in a circle is \(360^{\circ}\). Since there are 4 equal arcs (\(\overarc{XY},\overarc{YZ},\overarc{ZW},\overarc{WX}\)) in the circle for the inscribed square, \(m\overarc{XY}=\frac{360^{\circ}}{4} = 90^{\circ}\)

Step3: Measure of arc \(WY\)

Arc \(WY\) is composed of two arcs (\(\overarc{WX}\) and \(\overarc{XY}\)). Since each arc (like \(\overarc{XY}\)) is \(90^{\circ}\), \(m\overarc{WY}=90^{\circ}+90^{\circ}=180^{\circ}\)

Step4: Measure of \(\angle ZWX\)

\(\angle ZWX\) is an angle of the square. By the property of a square, all angles of a square are \(90^{\circ}\), so \(m\angle ZWX = 90^{\circ}\)

Answer:

\(m\overarc{XY}=90\), \(m\overarc{WY}=180\), \(m\angle ZWX = 90\)