Circle Geometry Theorems and Applications

Sep 27, 2024

Circle Geometry Theorems Lecture Notes

Introduction

  • Discussion on circle geometry theorems.
  • Six examples provided to practice and apply theorems.
  • Pause video to try examples before viewing solutions.

Key Theorems

  1. Center Theorems:
    • Angle at the center = 2 × angle at circumference.
    • Line from center perpendicular to chord bisects the chord.
    • Angle in a semicircle is 90 degrees.
  2. Cyclic Quadrilateral Theorems:
    • Opposite angles sum to 180 degrees.
    • Angles in the same segment are equal.
    • Exterior angle of a cyclic quad = sum of opposite interior angles.
    • Equal chords subtend equal angles.
  3. Tangent Theorems:
    • Tangent-chord angle = angle subtended by chord.
    • Two tangents from a point are equal in length.
    • Radius to tangent is perpendicular.

Example 1

  • Given: Circle with points A, B, C; angle A = 40°, angle B = 15°.
  • Find: Angles X, Y, Z.
    • Angle X (at center) = 2 × angle A = 80°.
    • Angles OBC and OCB (isosceles triangle) = 50° each (180° - 80° / 2).
    • Angle Z = 25° (180° - (40° + 15° + 50° + 50°)).

Example 2

  • Given: Cyclic quad ABCD.
  • Find: Lowercase a and b.
    • Lowercase a = 100°, lowercase b = 60° (opposite angles sum to 180°).

Example 3

  • Given: Circle with points L, M, N, K; tangent PMT; angle LMN = 35°.
  • Find: Angles A, B, C, D, E.
    • Angle A = 35° (tangent-chord theorem).
    • Angle B = 90° (diameter subtends 90°).
    • Angle C = 90° (tangent and radius perpendicular).
    • Angle D = 55° (angles on a straight line).
    • Angle E = 125° (opposite angles in cyclic quad).

Example 4

  • Given: Circle with center O; angles and parallel lines.
  • Find: Angles O1, Y2, and O2.
    • Angle O1 = 80° (at center).
    • Angles Y2 and Oly = 50° each (isosceles triangle).
    • Angle O2 = 20° (using parallel lines).

Example 5

  • Given: Circle with points A, B, C, D; tangent RDS; angle D3 = 10°.
  • Find: Angles B2, O1, AC, B1, and D1.
    • Angle B2 = 10° (opposite equal sides).
    • Angle O1 = 160° (sum of triangle angles).
    • Angle A = 80° (angle at center).
    • Angle C = 100° (opposite angles in cyclic quad).
    • Angles B1 and D2 = 40° (isosceles triangle).
    • Angle D1 = 40° (angle between tangent and chord).

Example 6

  • Given: Circle with center O; tangent line SUD; angle ATC = 105°, angle CTU = 40°.
  • Find: Angles A1, O1, and B2.
    • Angle O1 = 80° (angle at center).
    • Angle A1 = 40° (equal chords).
    • Angle B2 = 25° (sum of angles in cyclic quad).

Conclusion

  • Importance of understanding theorems and their applications.
  • Acknowledgment of different reasoning methods leading to the same answer.