Question

Triangle XYZ is inscribed inside a circle with diameter XZ and the inscribed angle at Y. The angle measure at X is 42 degrees.



Using what you know about inscribed angles, find the arc measure

Answer 1

In a circle, an inscribed angle is half the measure of the intercepted arc. In triangle XYZ, since XZ is the diameter of the circle, the angle at Y (∠Y) is a right angle (90 degrees) because it intercepts the semicircle.

We are given that the angle measure at X (∠X) is 42 degrees.

Now, to find the angle at Z (∠Z), we can use the triangle sum theorem, which states that the sum of the angles in any triangle is 180 degrees. Thus, we can find ∠Z as follows:

\[
\angle Y + \angle X + \angle Z = 180^\circ
\]
\[
90^\circ + 42^\circ + \angle Z = 180^\circ
\]
\[
\angle Z = 180^\circ - 132^\circ = 48^\circ
\]

Now that we have the angles in triangle XYZ, we can find the arc measures.

### Step 1: Find the measures of arcs
1. The measure of the arc that corresponds to angle X (∠X = 42°) is:
\[
\text{Arc YZ} = 2 \cdot \angle X = 2 \cdot 42^\circ = 84^\circ
\]

2. The measure of the arc that corresponds to angle Z (∠Z = 48°) is:
\[
\text{Arc XY} = 2 \cdot \angle Z = 2 \cdot 48^\circ = 96^\circ
\]

3. The remaining arc, which is arc XZ, is the remaining part of the circle:
\[
\text{Arc XZ} = 360^\circ - \text{Arc YZ} - \text{Arc XY}
\]
\[
= 360^\circ - 84^\circ - 96^\circ = 180^\circ
\]

### Conclusion
- The measure of arc YZ is \(84^\circ\).
- The measure of arc XY is \(96^\circ\).
- The measure of arc XZ (the arc intercepted by angle Y) is \(180^\circ\).

Hence, the measure of the arc corresponding to angle X is \(84^\circ\).