Integrated Math 2 / Unit 8: Circles
8.4 Angles Formed by Chords, Secants and Tangents
Notes
The angle's vertex position decides the formula.
- Inside the circle (two chords): angle $\displaystyle = \frac{1}{2}(\text{arc} + \text{vertical arc})$.
- Outside the circle (secants or tangents): angle $\displaystyle = \frac{1}{2}(\text{far arc} - \text{near arc})$.
- Tangent-chord angle (vertex on the circle): angle $\displaystyle = \frac{1}{2}$ the intercepted arc.
- Example: chords forming arcs of $80°$ and $100°$ meet at an angle of $\displaystyle \frac{1}{2}(80 + 100) = 90°$.
Practice
Two chords intersect inside a circle. The angle and its vertical angle intercept arcs of $80°$ and $100°$. What is the angle?
Inside angle $\displaystyle = \frac{1}{2}(80 + 100) = 90°$.
Two secants meet outside a circle and intercept a far arc of $120°$ and a near arc of $40°$. What is the angle?
Outside angle $\displaystyle = \frac{1}{2}(120 - 40) = 40°$.
A tangent and a chord meet at a point on the circle. The chord intercepts an arc of $150°$. What is the angle between them?
A tangent-chord angle is half its intercepted arc.
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