Question

in the figure, quadrilateral wxyz is inscribed in circle c. diagonals (overline{wy}) and (overline{xz}) intersect at c, the center of the circle.
*not drawn to scale
which statements are sufficient to conclude (angle xcy) must be a right angle?
choose all sufficient statements.
(square) a. (mangle xzy = 45^circ) (square) b. (mangle xyz = 90^circ) (square) c. (moverarc{xy} = 90^circ) (square) d. (moverarc{wzy} = 180^circ) (square) e. (moverarc{wx} = moverarc{xy}) (square) f. (moverarc{xy} = moverarc{zw})

Explanation:

To solve this, we analyze each option using circle theorems (central angles, inscribed angles, properties of cyclic quadrilaterals, etc.):

Option A: \( m\angle XZY = 45^\circ \)

\( \angle XZY \) is an inscribed angle subtended by arc \( XY \). The central angle \( \angle XCY \) subtended by the same arc \( XY \) is twice the inscribed angle:
\( m\angle XCY = 2 \times m\angle XZY \).
If \( m\angle XZY = 45^\circ \), then \( m\angle XCY = 2 \times 45^\circ = 90^\circ \). Thus, \( \angle XCY \) is a right angle.

Option B: \( m\angle XYZ = 90^\circ \)

\( \angle XYZ \) is an inscribed angle. A \( 90^\circ \) inscribed angle subtends a semicircle (arc \( XZ \) would be \( 180^\circ \)), but this tells us nothing about arc \( XY \) (the arc subtended by \( \angle XCY \)). Thus, this does not guarantee \( \angle XCY = 90^\circ \).

Option C: \( m\overarc{XY} = 90^\circ \)

The central angle \( \angle XCY \) subtends arc \( XY \). By definition, the measure of a central angle equals the measure of its subtended arc. Thus, if \( m\overarc{XY} = 90^\circ \), then \( m\angle XCY = 90^\circ \), so \( \angle XCY \) is a right angle.

Option D: \( m\overarc{WZY} = 180^\circ \)

Arc \( WZY \) is a semicircle (since its measure is \( 180^\circ \)), meaning \( WY \) is a diameter. However, this does not directly relate to arc \( XY \) or \( \angle XCY \). Thus, this does not guarantee \( \angle XCY = 90^\circ \).

Option E: \( m\overarc{WX} = m\overarc{XY} \)

Since \( WY \) is a diameter (diagonals intersect at the center, so \( WY \) is a diameter, making arc \( WY = 180^\circ \)), if \( m\overarc{WX} = m\overarc{XY} \), then each arc is \( \frac{180^\circ}{2} = 90^\circ \). Thus, \( m\overarc{XY} = 90^\circ \), so \( m\angle XCY = 90^\circ \) (by the central angle theorem).

Option F: \( m\overarc{XY} = m\overarc{ZW} \)

This only tells us the arcs \( XY \) and \( ZW \) are equal, but it does not specify their measure. Without knowing the measure of \( \overarc{XY} \), we cannot conclude \( \angle XCY = 90^\circ \).

Sufficient Statements:

A, C, and E are sufficient.

Answer:

A. \( m\angle XZY = 45^\circ \)
C. \( m\overarc{XY} = 90^\circ \)
E. \( m\overarc{WX} = m\overarc{XY} \)