Circles create many angle relationships when chords, secants, and tangents meet. These relationships are useful because they connect angle measures to the measures of intercepted arcs. Instead of measuring every angle directly, you can use arc information to calculate unknown angles.
The key idea is that the vertex location determines which formula applies.
When the vertex is on the circle, the angle is half its intercepted arc. When the vertex is inside the circle, the angle is half the sum of the intercepted arcs. When the vertex is outside the circle, the angle is half the difference of the intercepted arcs.
These rules appear in problems with intersecting chords, two secants, a secant and tangent, or two tangents.
Key Facts
- Inscribed angle: m∠ = 1/2(intercepted arc)
- Tangent-chord angle: m∠ = 1/2(intercepted arc)
- Two chords intersect inside: m∠ = 1/2(arc 1 + arc 2)
- Two secants intersect outside: m∠ = 1/2(larger arc - smaller arc)
- Tangent-secant angle outside: m∠ = 1/2(larger arc - smaller arc)
- Two tangents from the same outside point: m∠ = 1/2(major arc - minor arc) = 180° - minor arc
Vocabulary
- Chord
- A chord is a segment whose endpoints both lie on a circle.
- Secant
- A secant is a line that intersects a circle at two points.
- Tangent
- A tangent is a line that touches a circle at exactly one point.
- Intercepted arc
- An intercepted arc is the part of the circle cut off by the sides of an angle.
- Vertex location
- Vertex location describes whether the angle's vertex is inside the circle, on the circle, or outside the circle.
Common Mistakes to Avoid
- Using the same formula for every circle angle is wrong because the correct rule depends on whether the vertex is inside, on, or outside the circle.
- Forgetting the factor of 1/2 is wrong because these angle measures are based on half of an arc sum, arc difference, or intercepted arc.
- Adding arcs for an outside angle is wrong because angles formed outside a circle use half the difference of the intercepted arcs.
- Using the minor arc when the formula requires the larger arc is wrong because outside secant and tangent formulas depend on larger arc minus smaller arc.
Practice Questions
- 1 Two chords intersect inside a circle. The intercepted arcs are 86° and 134°. Find the measure of the angle formed.
- 2 Two secants intersect outside a circle. The larger intercepted arc is 210° and the smaller intercepted arc is 70°. Find the measure of the outside angle.
- 3 A tangent and a chord meet at a point on a circle. Explain why the angle formed is half the measure of its intercepted arc, and state which vertex location rule applies.