Question

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)

Answer 1

In a circle, the measure of an inscribed angle is half the measure of the arc that it subtends.

Since triangle XYZ is inscribed in the circle with diameter XZ, the angle at Y (∠Y) is an inscribed angle that subtends the arc XY.

Given that the measure of angle X (∠X) is 59°, we can find the measure of the arc XZ, which subtends this angle. Since XZ is a diameter of the circle, the arc XZ is 180°.

Now, the arc measure of XY can be calculated as follows:

1. The measure of angle X (∠X) is equal to the inscribed angle formed by the points X, Y, and Z, which is half the measure of the arc that it subtends. The inscribed angle at Y, which subtends arc XY, would thus be:
\[
\text{arc XY} = 2 \times (\text{measure of angle Y}) = 2 \times \text{measure of angle X}
\]

Since angle Y subtends the arc XY and ∠X is given as 59°, the arc XZ or the entire arc would measure as follows:

\[
\text{arc measure of XY} = 180° - \text{arc measure of XZ}
\]

Given that arc XZ measures 180°, we conclude that:

\[
\text{arc XY} = 180° - 59°
\]

That means:

\[
\text{arc XY} = 121°
\]

Thus, the measure of arc XY is 121°.