Theorem : 1 Angle at the Center is Twice the Angle at the Circumference

Theorem :
The angle subtended at the center of a circle by an arc is twice the angle subtended by the same arc at any point on the circumference of the circle.

Learning Objective:

• Understand the relationship between angles at the center and at the circumference
• Apply this theorem to find unknown angles in circle geometry
• Use the theorem in reasoning and geometric proofs

Definition / Concept

The angle subtended at the center of a circle by an arc is twice the angle subtended by the same arc at any point on the circumference of the circle.
$\text{If arc } AB \text{ subtends angle } \angle AOB \text{ at the center, and } \angle ACB \text{ at the circumference, then:}$

$∠ A O B = 2 ∠ A C B$ This holds for all points $C$ on the same side of the chord $AB$ (i.e., in the same segment).

Interactive Graph

Examples -1

In the diagram, $C$ is the center of the circle. The triangle is inscribed in the circle, and the angle at the circumference is $28^{\circ}$. The angle $m$ is subtended at the center by the same arc, and the angle $n$ is formed between a tangent and a chord.

Find the values of $m$ and $n$.

Answers ${m = 136^\circ, \quad n = 140^\circ}$

Examples -2

In the diagram, $C$ is the center of the circle. The triangle is isosceles, with two equal radii connecting the center to the circumference. The angle $x$ is at the center of the circle, and the angle $y$ is at the circumference, subtended by the same arc.

Find $y$ in terms of $x$.