Proving the Equality of Inscribed Angles in the Same Segment of a Circle

Understanding the properties of circles and their segments is fundamental in geometry. One of the essential properties involves the equality of inscribed angles subtended by the same arc. This article will illustrate a step-by-step proof that the angles on the same segment of a circle are equal, by utilizing the concept of inscribed angles and central angles.

Theorem

The angles subtended by the same arc at points on the circumference of a circle are equal.

Proof

Consider a circle with center O and points A, B, and C on the circumference of the circle such that the arc AB is subtended by angles at points C and D on the circumference.

1. Inscribed Angles

We need to show that the inscribed angles ∠ACB and ∠ADB are equal, where C and D are points on the circumference of the circle.

2. Draw Radii

Draw radii OA, OB, OC, and OD. These radii will form two triangles: ΔOAC and ΔOBD.

3. Central Angles

Let ∠AOB be the central angle subtended by the arc AB. According to the properties of a circle, the inscribed angle ∠ACB is half the measure of the central angle ∠AOB. Thus we have:

∠ACB 1/2 ∠AOB

Similarly for angle ∠ADB:

∠ADB 1/2 ∠AOB

4. Conclusion

Since both angles ∠ACB and ∠ADB are equal to 1/2 ∠AOB, we can conclude that:

∠ACB ∠ADB

Summary

The angles subtended by the same arc at any point on the circumference of the circle are equal. This property is fundamental in circle geometry and is often used in various geometric proofs and constructions.

Derivation of Angles in the Same Segment

Let O be the center of the circle. Let ADCB be a segment of the circle. For our convenience, let it be the major segment. Join AB. AB is a chord. Let ∠ADB and ∠ACB be the angles in the same segment. Join OA and OB.

∠AOB 2 × ∠ADB (angle subtended by the minor segment is twice the angle subtended by it in the remaining part)

Similarly, ∠AOB 2 × ∠ACB.

From the above two statements, it is clear that ∠ADB ∠ACB.

Alternative Proof Using Different Points on the Arc

Let AB be an arc of a circle. Let C be the center of the circle. Now join the ends of the arc to the center of the circle. The angle ACB subtended by arc AB at the center will be greater than 180°, equal to 180°, or less than 180° when the length of the arc AB is less than, equal to, or greater than half the circumference of the circle.

Now, let O be a point on the arc and join the ends of the arc to O. Now ∠AOB will always be half the angle ACB. Since O can be chosen anywhere on the arc AB, all angles AOB on an arc AB will be equal since every one of these angles AOB will be half of the angle ACB.