In a circle, the measure of an inscribed angle is half the measure of the intercepted arc. In this case, angle \( Y \) is an inscribed angle that intercepts arc \( XZ \).
Since \( XY \) is also an arc created by the points \( X \) and \( Y \), we can use the information we have:
- The inscribed angle at \( X \) has a measure of \( 59^\circ \).
- The arc opposite to angle \( X \) is the arc \( YZ \), which corresponds to angle \( Z \).
Since the triangle is inscribed in a circle, the angles in triangle \( XYZ \) add up to \( 180^\circ \): \[ \angle X + \angle Y + \angle Z = 180^\circ \] Let's denote the measure of angle \( Y \) as \( y \) and the measure of angle \( Z \) as \( z \). We already have: \[ 59^\circ + y + z = 180^\circ \] This simplifies to: \[ y + z = 121^\circ \]
Now, since arc \( XZ \) is opposite to angle \( Y \) and angle \( Y \) is an inscribed angle that intercepts arc \( XZ \), the relationship is: \[ \text{Arc } XZ = 2y \]
Also, since \( Y \) intercepts arc \( XZ \), \( XZ \) is the total arc for angles \( Y \) and \( Z \): \[ \text{Arc } XZ = \text{Arc } XY + \text{Arc } YZ \]
Now, in the triangle,
- Arc \( XY \) is opposite to angle \( Z \).
- Arc \( YZ \) is opposite to angle \( X \).
Let’s express these relationships in terms of angles:
- The measure of arc \( YZ = 2 \cdot \angle X = 2 \cdot 59^\circ = 118^\circ \).
- Therefore, we have: \[ \text{Arc } XZ = 360^\circ - \text{Arc } YZ \]
Now adding the contributions from both triangles: Arc \( XZ = \text{Arc } XY + \text{Arc } YZ\)
Therefore, \[ \text{Arc } XY = 118^\circ \]
Using the angle measures we determined, \[ y + z = 121^\circ \]
To find the arc measure of arc \( XY \), Calculate: \( \text{Arc } XY = 2 \cdot \text{Angle} Z\).
Since Angle Z is \( (121 - Y) \), replace this back in our equation of arc: \[ Arc XY = x \]
Through these relationships, we can derive the arc measures:
- Thus, the final measure of arc \( XY \) is \( 121^\circ - 59^\circ = 62^\circ\).
So, the arc measure of \( XY \) is: \[ \text{Arc } XY = 62^\circ \]
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