Inscribed angles are useful to proving certain characteristic of a circle. Memorizing the formulas of an inscribed angle will be useful in geometry.

**Inscribed Angle: **An angle where the vertex is on the circle’s circumference and the sides contain chords.

**Intercepted Arc: **The arc that is on the interior of the inscribed angle and whose endpoints are on the angle.

**Inscribed Polygon: **A polygon inside the circle with all the vertices on the circle’s circumference.

**Circumscribed Circle: **A circle that contains all the vertices of a polygon.

The vertex of an** inscribed angle** can be anywhere on the circle as long as its sides intersect the circle. The arc that is formed on the interior of the inscribed angle is the i**ntercepted arc.**

The center of the circle can be on the side of the inscribed angle, or it can be either inside or outside the angle.

*Theorem:* The measure of an inscribed angle is half the measure of its intercepted arc.

- \(m \angle ABC = \frac{1}{2}m\overset{ \huge\frown}{AC}\)

*Theorem: *Inscribed angles that intercept the same arc are congruent.

- \(\overset{ \huge\frown}{AB}\) is shared by ∠ADB and ∠BCA, so ∠ADB \(\cong\) ∠BCA

*Theorem: *An angle that intercepts a semicircle is a right angle.

- A right angle that inscribes a circle is a diameter.

When all the vertices of a polygon lie on a circle, the polygon is an **inscribed polygon**. The circle containing the vertices is a** circumscribed circle.**

*Theorem: *If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle.

- The converse is also true: If one side of an inscribed triangle is a diameter, then the triangle is a right triangle.

- m ∠DAB = 90° if and only if \(\overline{DB}\) is the diameter.

Not all quadrilaterals can be inscribed in a circle. Quadrilaterals that can be inscribed in a circle are called cyclic quadrilaterals

*Theorem:* A quadrilateral is inscribed in a circle if and only if the opposite angles are supplementary.

- Opposite angles (such as ∠A and ∠C in picture below) of an inscribed quadrilateral add up to 180°
- m ∠A + m ∠C = m ∠B + m ∠D = 180°
- Therefore, the quadrilateral ABCD is inscribed in the circle.