Circle Theorems: Angles and Properties

• Circle theorems describe angle relationships within circles
• These theorems dictate how angles at the center, circumference, and within cyclic quadrilaterals relate to each other

Angle at the Center Theorem

• The angle at the center of a circle subtended by an arc is twice the angle at the circumference subtended by the same arc
• If ∠AOB is the angle at the center and ∠ACB is the angle at the circumference, then ∠AOB = 2 × ∠ACB

Angles in the Same Segment Theorem

• Angles subtended by the same arc (or chord) in the same segment of a circle are equal
• If points A, B, C, and D lie on the circumference and angles ∠ACB and ∠ADB are subtended by arc AB, then ∠ACB = ∠ADB

Angle in a Semicircle Theorem

• The angle in a semicircle is always a right angle (90°)
• If AB is the diameter of the circle and C is any point on the circumference, then ∠ACB = 90°

Cyclic Quadrilateral Theorem

• A cyclic quadrilateral is a quadrilateral whose vertices all lie on the circumference of a circle
• The opposite angles of a cyclic quadrilateral are supplementary (add up to 180°)
• If ABCD is a cyclic quadrilateral, then ∠A + ∠C = 180° and ∠B + ∠D = 180°

Tangent-Radius Theorem

• A tangent to a circle is perpendicular to the radius drawn to the point of tangency
• If line PT is a tangent to the circle at point P and O is the center, then ∠OPT = 90°

Alternate Segment Theorem

• The angle between a tangent and a chord is equal to the angle in the alternate segment
• If PT is a tangent to the circle at point P and PQ is a chord, then the angle between the tangent and chord (∠TPQ) is equal to the angle subtended by the chord in the alternate segment (∠PRQ)

Corollaries and Extensions

• Equal chords subtend equal angles at the center
• Chords equidistant from the center are equal in length
• The line from the center of a circle to the midpoint of a chord is perpendicular to the chord
• The perpendicular bisector of a chord passes through the center of the circle

Problem Solving Techniques

• Identify the relevant circle theorem(s) applicable to the problem
• Look for key elements such as center, circumference, tangents, chords, and cyclic quadrilaterals
• Apply the theorem(s) to set up equations involving unknown angles
• Solve for the unknown angles using algebraic techniques
• Provide justification for each step based on circle theorems

Common Geometric Terms

• Circle: A set of points equidistant from a center point
• Center: The point equidistant from all points on a circle
• Radius: The distance from the center to any point on the circle
• Diameter: A chord passing through the center of the circle (twice the radius)
• Chord: A line segment joining two points on a circle
• Arc: A portion of the circumference of a circle
• Tangent: A line that touches the circle at only one point
• Secant: A line that intersects the circle at two points
• Circumference: The distance around the circle
• Segment: The region bounded by a chord and an arc
• Sector: The region bounded by two radii and an arc
• Subtend: An angle is subtended by an arc or chord if its rays pass through the endpoints of the arc or chord

Properties of Chords

• A line segment connecting two points on a circle's circumference is called a chord
• A diameter is the longest chord in a circle, passing through the center
• If a radius bisects a chord, it is perpendicular to that chord
• Conversely, a radius that is perpendicular to a chord bisects that chord
• Equal chords in a circle subtend equal angles at the center of the circle
• Parallel chords intercept equal arcs on the circumference

Applications

• Circle theorems are used extensively in geometry and trigonometry
• Used in engineering to design circular structures and mechanisms
• Used in architecture for designing arches and domes
• Used in navigation and astronomy for calculating angles and distances on a sphere