Angles Formed by Intersecting Chords
When two chords intersect inside a circle, each angle formed equals half the sum of the two intercepted arcs.
Introduction
Chords AC and BD intersect at P: angle = ½(arc a + arc b)
Past Knowledge
Inscribed angles (10.1.4). Arc addition (10.1.3). Vertical angles.
Today's Goal
Apply the intersecting chords angle formula.
Future Success
Secant angles (10.2.3), chord segment products (10.2.4).
Key Concepts
Intersecting Chords Angle Theorem
When two chords intersect inside a circle:
where arc₁ and arc₂ are the arcs intercepted by the angle and its vertical angle.
Memory Aid: “Inside = Average”
When chords cross inside a circle, the angle is the average of the two arcs it “sees”. Half the sum = average.
Theorem & Proof
Proof: Intersecting Chords Angle = ½(sum of intercepted arcs)
Given: Chords and intersect at inside circle
Prove:
| # | Statement | Reason |
|---|---|---|
| 1 | Draw auxiliary chord | Two points determine a line |
| 2 | Inscribed Angle Theorem | |
| 3 | Inscribed Angle Theorem | |
| 4 | Exterior Angle Theorem (∠APB is exterior to △DPC) |
∎ Drawing the auxiliary chord creates inscribed angles, and the exterior angle theorem combines them.
Worked Examples
Finding the Angle
Two chords intersect inside a circle. Intercepted arcs are 80° and 40°. Find the angle.
60°
Finding a Missing Arc
Chords intersect at 75°. One intercepted arc is 100°. Find the other.
→ →
50°
All Four Arcs
Chords create arcs of (3x)°, (x + 20)°, (5x − 10)°, and (2x)°. Find x and all arcs.
Sum = 360°:
→
Actually: → ≈ 31.8. Let me use cleaner numbers.
Better: arcs (4x)°, (2x)°, (3x + 20)°, (x + 20)° sum to 360°: →
Arcs: 128°, 64°, 116°, 52°. Angle between arcs 128° and 116°:
x = 32; arcs: 128°, 64°, 116°, 52°
Common Pitfalls
Using the Wrong Arcs
Use the arc the angle “looks at” and its vertical angle's arc. Not the adjacent arcs.
Applying the Wrong Formula
Inside = ½(sum). Outside = ½(difference). Check where the vertex is!
Real-Life Applications
Wheel Spoke Intersections
Bicycle wheels with crossing spokes create intersecting chords. The angles at each crossing point follow the intersecting chords theorem.
Camera Optics
Light rays crossing inside a circular lens create intersecting chord patterns, and the refraction angles depend on the arcs intercepted.
Practice Quiz
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