Overview
This lecture covers the different types of angles related to circles and how to calculate angle and arc measures using key rules and formulas.
Central Angles
- A central angle has its vertex at the center of the circle.
- The measure of a central angle equals the measure of its intercepted arc.
Inscribed Angles
- An inscribed angle has its vertex on the circle and is formed by two chords.
- The inscribed angle is half the measure of its intercepted arc.
- If a chord passes through the center, it's called a diameter; the inscribed angle across a diameter is always 90°.
Tangent-Chord Angles
- A tangent-chord angle is formed by a chord and a tangent meeting at one point.
- This angle is half the measure of its intercepted arc; the arc is twice the angle.
Chord-Chord Angles
- Formed by intersecting chords inside the circle.
- The measure of the angle is half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
Secant-Secant, Secant-Tangent, and Tangent-Tangent Angles
- Secant-secant and secant-tangent angles' measure is half the difference between the intercepted arcs.
- Tangent-tangent angle is half the difference between the major and minor intercepted arcs.
Problem-Solving Strategies
- When given angles or arcs, use algebraic equations based on the angle rules above.
- The sum of all arcs in a circle is 360°.
- When angles form a linear pair, their sum is 180°.
- In triangle problems, the sum of interior angles is 180°.
Key Terms & Definitions
- Central angle — Angle with vertex at the circle's center.
- Inscribed angle — Angle with vertex on the circle, formed by two chords.
- Chord — Line segment connecting two points on a circle.
- Tangent — Line touching the circle at exactly one point.
- Secant — Line that crosses the circle at two points.
- Intercepted arc — Arc corresponding to a given angle.
- Chord-chord angle — Angle from two intersecting chords, measured by half the sum of intercepted arcs.
Action Items / Next Steps
- Practice problems calculating arc and angle measures for each angle type.
- Review definitions and formulas for each circle angle.
- Complete any assigned homework or reading on circle geometry.