Question

in the figure, quadrilateral wxyz is inscribed in circle c. diagonals wy and xz intersect at c, the center of the circle.
*not drawn to scale
which statements are sufficient to conclude ∠xcy must be a right angle?
choose all sufficient statements.
a. m∠xzy = 45°

b. m∠xyz = 90°

c. mwidehat{xy} = 90°

d. mwidehat{wzy} = 180°

e. mwidehat{wx} = mwidehat{xy}

f. mwidehat{xy} = mwidehat{zw}

Explanation:

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

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

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

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

\( \angle XYZ \) is an inscribed angle. A \( 90^\circ \) inscribed angle subtends a semicircle, so arc \( XZ \) is \( 180^\circ \) (diameter). But this tells us about arc \( XZ \), not arc \( XY \) (which determines \( \angle XCY \)). \( \angle XCY \) depends on arc \( XY \), not \( XZ \). Not sufficient.

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

The central angle \( \angle XCY \) subtends arc \( XY \). By definition, the measure of a central angle equals the measure of its subtended arc. If \( m\widehat{XY} = 90^\circ \), then \( m\angle XCY = 90^\circ \). Sufficient.

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

Arc \( WZY \) being \( 180^\circ \) means \( WY \) is a diameter (since a semicircle has measure \( 180^\circ \)). But this tells us \( WY \) is a diameter, not about arc \( XY \) (or \( \angle XCY \)). Not sufficient.

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

This means arcs \( WX \) and \( XY \) are congruent, but it does not fix their measure to \( 90^\circ \). For example, they could each be \( 60^\circ \), so \( \angle XCY \) would be \( 60^\circ \), not \( 90^\circ \). Not sufficient.

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

This states arcs \( XY \) and \( ZW \) are congruent, but does not fix their measure to \( 90^\circ \). For example, they could each be \( 80^\circ \), so \( \angle XCY = 80^\circ \). Not sufficient.

Answer:

A. \( m\angle XZY = 45^\circ \), C. \( m\widehat{XY} = 90^\circ \)