Answer:
66°
Explanation:
You want the measure of arc YZ in semicircle XYZ with inscribed angle XZY given as 57°.
Inscribed angle
The measure of an inscribed angle is half the measure of the arc it subtends. This means arc XY is double the measure of inscribed angle XZY.
arc XY = 2×∠XZY = 2×57° = 114°
Semicircle
The measure of semi-circular arc XYZ is 180°, and is the total of the measures of arcs XY and YZ.
arc XY +arc YZ = 180°
arc YZ = 180° -arc XY = 180° -114°
arc YZ = 66°
The measure of the indicated arc is 66°.
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Additional comment
∆XYZ is a right triangle, so angle X is the complement of angle Z: 33°. Arc YZ is double the measure of angle YXZ, or 66°.
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