Triangle XYZ is inscribed inside a circle with diameter XZ and the inscribed angle at Y . The angle measure at X is 59°. Using what you know about inscribed angles, find the arc measure of XY .(1 point)

Answer 1

In a circle, an inscribed angle is half the measure of the arc it intercepts. In this case, triangle \( XYZ \) is inscribed in a circle with a diameter \( XZ \) and the inscribed angle at \( Y \) means that angle \( Y \) intercepts arc \( XZ \).

Since \( XZ \) is a diameter, the angle \( Y \) is a right angle (90°) due to the inscribed angle theorem, where any angle inscribed in a semicircle is a right angle.

We know that angle \( X \) measures \( 59° \). Therefore, angle \( Z \) can be computed using the triangle sum property, which states that the sum of angles in a triangle is \( 180° \). Thus:

\[ \text{Angle } Y + \text{Angle } X + \text{Angle } Z = 180° \]

Substituting \( Y = 90° \) and \( X = 59° \):

\[ 90° + 59° + \text{Angle } Z = 180° \] \[ \text{Angle } Z = 180° - 90° - 59° = 31° \]

Now, angle \( Z \) intercepts arc \( XY \). According to the inscribed angle theorem:

\[ \text{Arc } XY = 2 \times \text{Angle } Z \] \[ \text{Arc } XY = 2 \times 31° = 62° \]

Thus, the arc measure of \( XY \) is \( \boxed{62°} \).