Inscribed In Circles

Big Picture

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

Key Terms

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.

Inscribed Angles

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 intercepted arc.

Inscribed Angles

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

Inscribed Angles ARC

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

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

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

  • \(\overset{ \huge\frown}{AB}\) is shared by  ADB  and  BCA, so ADB \(\cong\) BCA
Inscribed Polygons

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

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


InscrIbed In CIrcles Cont.

Inscribed Polygons

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.

Inscribed Polygons

Inscribed Triangle

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

Inscribed Quadrilaterals

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.
Inscribed Quadrilaterals