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Theorems related to Angles in a Circle

Introduction

In Chapter 7, “Theorems Related to Angles in a Circle,” students explore fundamental properties of circles and the angles they harbour. This chapter forms a cornerstone for geometrical reasoning, enabling learners to understand relationships such as inscribed angles, central angles, and the way they interact with arcs. Through clear theorems and illustrative examples, students build a strong base in circle geometry—critical for both board exams and higher mathematics.

1. Central Angle Theorem

A central angle is an angle whose vertex is at the centre of the circle. The Central Angle Theorem states:

  • Statement: The measure of a central angle is equal to the measure of its intercepted arc. 
  • Explanation: If angle AOB is a central angle intercepting arc AB on circle with centre O, then ∠AOB = measure of arc AB. 
  • Example: In a circle with centre O, if arc AB measures 60°, then ∠AOB also measures 60°. 

2. Inscribed Angle Theorem

  • Statement: An inscribed angle is half the measure of its intercepted arc. 
  • Explanation: For an inscribed angle ∠APB intercepting arc AB, ∠APB = ½ × measure of arc AB. 
  • Example: If arc AB measures 80°, then the inscribed angle ∠APB equals 40°. 

3. Angle in a Semicircle

  • Statement: An angle inscribed in a semicircle is a right angle (90°). 
  • Explanation: If an inscribed angle intercepts a semicircular arc (180°), then it equals ½ × 180° = 90°. 
  • Example: In circle with diameter AB, any point P on the circumference yields ∠APB = 90°. 

4. Angles in the Same Segment

  • Statement: Angles drawn from the same segment of a circle are equal. 
  • Explanation: If two inscribed angles intercept the same arc (or chord), they are congruent. 
  • Example: In a circle, points A, B, C, and D lie on the circumference; if both ∠ACB and ∠ADB intercept arc AB, then ∠ACB = ∠ADB. 

5. Angle Between Tangent and Chord

 

  • Statement: The angle between a tangent and a chord through the point of contact is equal to the angle in the opposite segment. 
  • Explanation: If PT is a tangent at point T on the circle, and chord TA forms angle ∠PTA, then ∠PTA = angle in the opposite segment—i.e., angle subtended by chord TA in the alternate arc. 
  • Example: With tangent PT at T and chord TA intercepting circle at A and B, ∠PTA equals the angle subtended in the opposite arc, say ∠TBA. 

6. Cyclic Quadrilateral Angle Theorem

  • Statement: Opposite angles of a cyclic quadrilateral sum to 180°. 
  • Explanation: In quadrilateral ABCD inscribed in a circle, ∠A + ∠C = 180°, and ∠B + ∠D = 180°. 
  • Example: If ∠A = 70°, then ∠C = 110°, making opposite‑angle sum a straight angle. 

7. Example Problems (Expanded)

Here are expanded examples for clarity:

Example A – Central and Inscribed Angle Combined

Circle with centre O. Arc AB measures 120°.

  • Central angle ∠AOB = 120°. 
  • Choose a point C on the remaining circumference. Then inscribed angle ∠ACB intercepting arc AB = ½ × 120° = 60°. 
  • Interpretation: This illustrates the direct relationship in theorems—central vs. inscribed angle. 

Example B – Angle in Semicircle with Tangent

Let AB be diameter; point C is on circumference. Tangent at C is CT; chord CA.

  • By semicircle theorem: ∠ACB = 90°. 
  • By tangent‑chord theorem: ∠ACT = angle in opposite segment—i.e., ∠ABC.
     If ∠ABC = 35°, then ∠ACT = 35°. 

Example C – Cyclic Quadrilateral

Quadrilateral ABCD inscribed in a circle: ∠A = 65°, ∠B = 80°.

  • Then ∠C = 180° − 65° = 115°, and ∠D = 180° − 80° = 100°.
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