Inscribed Angles
An inscribed angle has its vertex on the circle. Its measure is always half the intercepted arc — one of the most important circle theorems.
Introduction
Inscribed angle θ = ½ intercepted arc (2θ)
Past Knowledge
Central angles (10.1.2). Arc addition (10.1.3). Triangle theorems.
Today's Goal
Prove and apply the Inscribed Angle Theorem.
Future Success
Cyclic quadrilaterals (10.1.5), chord angles (10.2.2).
Key Concepts
Inscribed Angle Theorem
Key Corollaries
- Same arc → same angle: All inscribed angles intercepting the same arc are congruent
- Semicircle → right angle: An inscribed angle that intercepts a semicircle (diameter) is exactly 90°
- Inscribed = ½ Central: An inscribed angle is half the central angle that intercepts the same arc
Theorem & Proof
Two-Column Proof: Inscribed Angle Theorem (Case 1: One Side Through Center)
Given: Inscribed with passing through center
Prove:
| # | Statement | Reason |
|---|---|---|
| 1 | All radii of a circle are congruent | |
| 2 | is isosceles | Two sides congruent (step 1) |
| 3 | Base angles of isosceles triangle are congruent | |
| 4 | Exterior Angle Theorem: | |
| 5 | Central angle = intercepted arc | |
| 6 | From steps 3 and 5: |
∎ The isosceles triangle formed by two radii, combined with the Exterior Angle Theorem, gives the factor of ½. The other two cases (center inside/outside the angle) follow by adding/subtracting this case.
Worked Examples
Arc → Inscribed Angle
Inscribed angle intercepts an arc of 140°. Find the angle.
70°
Angle in a Semicircle
Triangle ABC is inscribed in circle O with AB as a diameter. Find ∠ACB.
∠ACB intercepts semicircle AB = 180°.
∠ACB = 90° — Thales' Theorem!
Two Inscribed Angles
Inscribed ∠BAC = (3x + 5)° and inscribed ∠BDC = (5x − 15)° both intercept arc BC. Find x and the angles.
Same arc → same inscribed angle:
→
Each angle = 3(10) + 5 = 35°
∠BAC = ∠BDC = 35°
Common Pitfalls
Doubling When You Should Halve
Inscribed angle = ½ arc. Going from angle to arc: multiply by 2. Going from arc to angle: divide by 2. Getting the direction wrong is the #1 error.
Confusing Central and Inscribed
Central angle = arc (no division). Inscribed angle = ½ arc. Check where the vertex is: at the center or on the circle.
Real-Life Applications
Thales' Theorem in Construction
To construct a perfect right angle without a protractor, draw a semicircle and connect any point on it to the diameter's endpoints. Ancient builders used this technique to square foundations.
Satellite Dish Design
Parabolic satellite dishes use properties related to inscribed angles to focus incoming signals to a single point (the feed horn) at the dish's focal point.
Practice Quiz
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