Inscribed Quadrilaterals

Opposite angles in an inscribed quadrilateral always add up to 180°. Always.

Opposite angles in an inscribed quadrilateral always add up to 180°.

How It Works

What does inscribed mean?

Inscribed means the shape is inside the circle with all its corners (vertices) touching the circle. Think of it as the shape fitting perfectly inside.

The 180° rule

For any quadrilateral inscribed in a circle: opposite angles add up to 180°. If angle A = 70°, then the angle across from it = 110°. Because 70 + 110 = 180.

Using this to solve problems

If you know one angle, you can find the opposite angle right away. Just subtract from 180°. If angle B = 95°, angle D = 180° - 95° = 85°.

Connecting to arcs

Each inscribed angle = half its intercepted arc. So the arcs intercepted by opposite angles must add up to 360° (the whole circle). This helps when you know arc measures.

Key Formulas

Opposite Angles Rule

angle A + angle C = 180°

Opposite angles in an inscribed quadrilateral are supplementary (add to 180°).

Other Pair Too

angle B + angle D = 180°

Both pairs of opposite angles add to 180°, not just one pair.

Arc Connection

Each angle = ½ × (its intercepted arc)

Each angle in the inscribed quadrilateral is an inscribed angle, so it equals half its intercepted arc.