Question

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. m(widehat{xy})=90° d. m(widehat{wzy})=180° e. m(widehat{wx})=m(widehat{xy}) f. m(widehat{xy})=m(widehat{zw})

Explanation:

Step1: Recall central - inscribed angle relationship

The measure of an inscribed angle is half the measure of the central angle that subtends the same arc.

Step2: Analyze option A

If \(m\angle XZY = 45^{\circ}\), the central angle \(m\angle XCY\) that subtends arc \(XY\) is \(2m\angle XZY=90^{\circ}\) since the central - inscribed angle relationship (\(m\angle XCY = 2m\angle XZY\)).

Step3: Analyze option B

If \(m\angle XYZ = 90^{\circ}\), the arc \(XZ\) is a semi - circle. But this does not directly tell us about \(\angle XCY\).

Step4: Analyze option C

If \(m\overset{\frown}{XY}=90^{\circ}\), the central angle \(m\angle XCY\) which is the measure of the arc it subtends is \(90^{\circ}\) (the measure of a central angle is equal to the measure of its subtended arc).

Step5: Analyze option D

If \(m\overset{\frown}{WZY}=180^{\circ}\), it means \(WY\) is a diameter. But this does not directly imply that \(\angle XCY = 90^{\circ}\).

Step6: Analyze option E

If \(m\overset{\frown}{WX}=m\overset{\frown}{XY}\), we cannot conclude that \(\angle XCY = 90^{\circ}\) without more information.

Answer:

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