Geometry Test Questions: Angles in Semicircles and Inscribed Shapes

📚 Quick Study Guide

• 📐 Angle in a Semicircle: An angle inscribed in a semicircle is always a right angle (90 degrees).
• ⏺️ Inscribed Angle Theorem: The measure of an inscribed angle is half the measure of its intercepted arc.
• ⧖ Cyclic Quadrilateral: The opposite angles in a cyclic quadrilateral (a quadrilateral inscribed in a circle) are supplementary (add up to 180 degrees).
• 💡 Central Angle: The measure of a central angle is equal to the measure of its intercepted arc.
• 🔗 Relationship Between Central and Inscribed Angles: An inscribed angle is half of the central angle that subtends the same arc.

Practice Quiz

1. What is the measure of angle ABC if it is inscribed in a semicircle with AC as the diameter?
   1. 45°
   2. 60°
   3. 90°
   4. 120°
2. In circle O, angle AOC is a central angle measuring 80°. If point B lies on the circle such that angle ABC is an inscribed angle intercepting the same arc, what is the measure of angle ABC?
   1. 40°
   2. 60°
   3. 80°
   4. 160°
3. Quadrilateral ABCD is inscribed in a circle. If angle A measures 100°, what is the measure of angle C?
   1. 80°
   2. 90°
   3. 100°
   4. 180°
4. In a circle, arc AB measures 120°. What is the measure of the inscribed angle that intercepts arc AB?
   1. 30°
   2. 60°
   3. 120°
   4. 240°
5. If an inscribed angle in a circle measures 35°, what is the measure of the arc it intercepts?
   1. 35°
   2. 70°
   3. 105°
   4. 140°
6. In cyclic quadrilateral PQRS, if angle P = $2x + 10$ and angle R = $x + 20$, find the value of x.
   1. 30
   2. 50
   3. 60
   4. 90
7. A square is inscribed in a circle. What is the measure of the angle subtended by a side of the square at the center of the circle?
   1. 45°
   2. 60°
   3. 90°
   4. 120°