Question

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)
º

Answer 1

In a circle, the measure of an inscribed angle is half the measure of the arc that it intercepts. In triangle XYZ, angle Y is the inscribed angle that intercepts arc XY.

Given that the diameter XZ implies that angle Y is a right angle (90°) since any angle inscribed in a semicircle is a right angle.

Since you already know that angle X measures 59°, you can determine angle Z using the fact that the sum of the angles in a triangle is 180°:

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

\[
\text{Angle Y} + 59° + \text{Angle Z} = 180°
\]

Since angle Y is a right angle and measures 90°:

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

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

Now, we know that the measure of arc XY is twice the inscribed angle Y (because the inscribed angles are half the measure of the arcs they intercept).

Thus, to find the measure of arc XY:

\[
\text{Arc measure of } XY = 2 \times \text{Angle Y} = 2 \times 90° = 180°
\]

However, arc XY is actually intercepted by angle X, so we need to find the arc that corresponds to angle X, which is:

\[
\text{Arc measure of } XY = 2 \times \text{Angle X} = 2 \times 59° = 118°
\]

Therefore, the measure of arc XY is:

\[
\boxed{118°}
\]