circle theorems examples grade 10 math

📚 Quick Study Guide

• 📐 Angle at the Center Theorem: The angle subtended by an arc at the center of the circle is twice the angle subtended by it at any point on the remaining part of the circle. Mathematically, $\angle AOC = 2 \angle ABC$.
• ✍️ Angle in the Same Segment Theorem: Angles in the same segment of a circle are equal. Therefore, $\angle BAC = \angle BDC$.
• ➗ Angle in a Semicircle Theorem: The angle in a semicircle is always a right angle ($90^{\circ}$). If AC is the diameter, then $\angle ABC = 90^{\circ}$.
• 🔄 Cyclic Quadrilateral Theorem: The opposite angles of a cyclic quadrilateral add up to $180^{\circ}$. Thus, $\angle A + \angle C = 180^{\circ}$ and $\angle B + \angle D = 180^{\circ}$.
• 🎯 Tangent Radius Theorem: A tangent to a circle is perpendicular to the radius at the point of contact. So, $\angle OTB = 90^{\circ}$.

Practice Quiz

1. Question 1: In a circle with center O, angle AOC = 80°. What is the measure of angle ABC, where B is a point on the remaining part of the circle?
   1. A. 20°
   2. B. 40°
   3. C. 60°
2. Question 2: Points A, B, C, and D lie on the circumference of a circle. If angle BAC = 35°, what is the measure of angle BDC?
   1. A. 35°
   2. B. 55°
   3. C. 70°
3. Question 3: AC is the diameter of a circle, and B is a point on the circumference. What is the measure of angle ABC?
   1. A. 45°
   2. B. 60°
   3. C. 90°
4. Question 4: ABCD is a cyclic quadrilateral. If angle A = 100°, what is the measure of angle C?
   1. A. 60°
   2. B. 80°
   3. C. 100°
5. Question 5: A tangent TB touches a circle with center O at point B. If angle TBO = x, what is the value of x?
   1. A. 0°
   2. B. 45°
   3. C. 90°
6. Question 6: In a circle, if the angle at the centre is 150°, what is the angle at the circumference subtended by the same arc?
   1. A. 50°
   2. B. 75°
   3. C. 100°
7. Question 7: PQRS is a cyclic quadrilateral. If angle P = 70°, what is the measure of angle R?
   1. A. 70°
   2. B. 110°
   3. C. 140°