Inscribed Polygons & Cyclic Quadrilaterals
A polygon inscribed in a circle has all its vertices on the circle. For quadrilaterals, this creates a beautiful property: opposite angles are supplementary.
Introduction
Cyclic quadrilateral: ∠A + ∠C = 180°
Past Knowledge
Inscribed angles (10.1.4). Arc addition (10.1.3). Quadrilateral angle sum = 360°.
Today's Goal
Prove opposite angles in a cyclic quadrilateral are supplementary.
Future Success
Tangent lines (10.2.1), advanced circle theorems, SAT problems.
Key Concepts
Cyclic Quadrilateral Theorem
If a quadrilateral is inscribed in a circle (cyclic), then:
Converse
If a quadrilateral's opposite angles are supplementary, then it CAN be inscribed in a circle (it is cyclic).
Inscribed Right Triangles
A right triangle is always inscribed in a semicircle — the hypotenuse is a diameter (Thales' converse).
Theorem & Proof
Two-Column Proof: Opposite Angles of a Cyclic Quadrilateral Sum to 180°
Given: Quadrilateral inscribed in circle
Prove:
| # | Statement | Reason |
|---|---|---|
| 1 | Inscribed Angle Theorem (∠A intercepts arc BCD) | |
| 2 | Inscribed Angle Theorem (∠C intercepts arc BAD) | |
| 3 | Full circle = 360° | |
| 4 | Add steps 1 + 2 and substitute step 3 |
∎ Opposite inscribed angles intercept arcs that together make the full circle (360°), so the angles sum to half of 360° = 180°.
Worked Examples
Finding an Opposite Angle
Cyclic quadrilateral ABCD. ∠A = 75°. Find ∠C.
∠C = 105°
Algebraic
Cyclic ABCD: ∠A = (2x + 10)°, ∠C = (3x)°. Find all four angles.
→ →
∠A = 78°, ∠C = 102°. ∠B + ∠D = 180° also.
∠A = 78°, ∠C = 102°
Is It Cyclic?
A quadrilateral has angles 80°, 100°, 100°, 80°. Is it cyclic?
Check opposite pairs: 80° + 100° = 180° ✓ and 100° + 80° = 180° ✓
Yes — opposite angles are supplementary, so it can be inscribed in a circle.
Common Pitfalls
Adjacent ≠ Opposite
It's opposite angles that sum to 180°, not adjacent ones. In a cyclic quadrilateral ABCD, ∠A + ∠C = 180° (not ∠A + ∠B).
Not All Quadrilaterals Are Cyclic
A generic parallelogram is NOT cyclic (its opposite angles are equal, not supplementary — unless it's a rectangle, where 90° + 90° = 180°).
Real-Life Applications
Circumscribed Circles in Architecture
Gothic rose windows inscribe complex polygons in circles. The supplementary angle property ensures structural symmetry and even weight distribution.
Compass Navigation
The method of trilateration (used in GPS) relies on inscribed angle properties to determine positions relative to three known points on a circle.
Practice Quiz
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