Lesson 5 — Secant Angles
Big idea: A secant angle’s measure depends on where its vertex is — inside the circle use half the sum of two arcs; outside use half the difference.
Words to know
| Word | Meaning |
|---|---|
| Secant | A line that intersects a circle in two points. |
| Secant angle | An angle whose sides are contained in two secants, with each side meeting the circle in at least one point other than the vertex. |
| Intercepted arcs | The arcs cut off by the sides of the angle. |
| Vertical angles | The opposite angles formed by two intersecting lines; they are equal. |
| Vertex inside | The angle’s vertex is inside the circle. |
| Vertex outside | The angle’s vertex is outside the circle. |
Concepts to know
- A secant cuts through a circle at two points; a tangent touches at one.
- Theorem 64: A secant angle whose vertex is inside a circle equals half the sum of the arcs intercepted by it and its vertical angle.
- Theorem 65: A secant angle whose vertex is outside a circle equals half the difference of its larger and smaller intercepted arcs.
- For an inside vertex, the two relevant arcs are the one inside the angle and the one inside its vertical angle.
- For an outside vertex, the two relevant arcs are the far arc and the near arc.
Formulas / rules
- Vertex inside: angle = ½(arc₁ + arc₂)
- Vertex outside: angle = ½(larger arc − smaller arc)
- The two arcs in the inside case are intercepted by the angle and its vertical angle.
- The two arcs in the outside case are the two arcs between the sides of the angle.
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Find an angle with vertex inside | Add the two intercepted arcs, then divide by 2. |
| Find an angle with vertex outside | Subtract the smaller arc from the larger, then divide by 2. |
| Find an arc from an inside angle | Double the angle, then subtract the other arc. |
| Find an arc from an outside angle | Double the angle, then add the smaller arc. |
| Decide which theorem to use | Vertex at the center → central; on the circle → inscribed; inside → Theorem 64; outside → Theorem 65. |
| Work with two secants from an outside point | Use Theorem 65 with the far arc and the near arc. |
Watch out
- Inside means add the arcs; outside means subtract them. Mixing these up is the most common error.
- Outside: subtract the smaller arc from the larger so the angle is positive.
- Use the arcs actually cut off by the angle’s sides, not any arc in the figure.
- An inscribed angle has its vertex on the circle; use Theorem 63, not Theorem 64.
- Double-check whether the question gives the angle and asks for an arc — then reverse the formula.