Circles - Inscribed angles Worksheets - Free Printable

I'll solve this geometry problem step by step.

First, look closely:
- The image shows a circle with points X, Y, and Z on its circumference.
- There's an angle at point Z that we need to find (marked with a question mark).
- We're given two arc measures: arc YZ is 110° and arc XZ is 146°.
- This appears to be a problem about inscribed angles in a circle.

The key concept here is that an inscribed angle (an angle formed by two chords with a common endpoint on the circle) is half the measure of the intercepted arc. In this case, angle XZY (at point Z) intercepts arc XY.

To find the measure of angle XZY, I need to find the measure of arc XY first.

Since the total circumference of a circle is 360°, and we have arcs YZ (110°) and XZ (146°), I can find arc XY by subtracting these from 360°.

Let me calculate:
Arc XY = 360° - Arc YZ - Arc XZ
Arc XY = 360° - 110° - 146°
Arc XY = 360° - 256°
Arc XY = 104°

Now, since angle XZY is an inscribed angle intercepting arc XY, its measure is half of arc XY:
Angle XZY = 1/2 × Arc XY
Angle XZY = 1/2 × 104°
Angle XZY = 52°

Therefore, the answer is A) 52°.

This makes sense because the inscribed angle theorem states that an inscribed angle is half the measure of its intercepted arc, and we've correctly identified which arc is intercepted by the angle at Z.