Overview
This lesson covers various types of angles related to circles, including their definitions, properties, and methods for calculating arc and angle measures.
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 sides that are chords.
- The measure of an inscribed angle is half the measure of its intercepted arc.
Tangent-Chord Angles
- Formed where a tangent meets a chord at the point of tangency.
- The angle is half the measure of its intercepted arc; the arc is twice the angle.
Chord-Chord Angles
- Formed by the intersection of two chords inside a circle.
- The angle measure is half the sum of the measures of the intercepted arcs.
- To solve for an unknown arc or angle, set up an equation using these relationships and solve algebraically.
Secant-Secant Angles
- Formed outside a circle by the intersection of two secants.
- The angle is half the difference of the intercepted arcs.
Secant-Tangent Angles
- Formed outside a circle by the intersection of a secant and a tangent.
- The angle is half the difference of the intercepted arcs.
Tangent-Tangent Angles
- Formed outside a circle by the intersection of two tangents.
- The angle is half the difference between the major and minor arcs they intercept.
Solving Problems & Relationships
- The sum of arcs around a circle is 360°.
- In triangles formed by diameters, the angle opposite the diameter is always 90°.
- For inscribed angles that share the same arc, the angles are congruent.
- Use algebraic equations to solve for unknowns in problems involving arc and angle measures.
Key Terms & Definitions
- Central Angle — Angle with vertex at the center of the circle.
- Inscribed Angle — Angle with vertex on the circle and sides as chords.
- Chord — Line segment connecting two points on a circle.
- Diameter — Chord passing through the center of the circle.
- Tangent — Line touching the circle at exactly one point.
- Secant — Line or segment intersecting the circle at two points.
- Intercepted Arc — Arc that lies in the interior of an angle.
Action Items / Next Steps
- Practice identifying and calculating measures for central, inscribed, tangent-chord, chord-chord, secant-secant, secant-tangent, and tangent-tangent angles.
- Complete assigned exercises involving finding unknown angles and arcs using the discussed formulas.