Circle Theorems: Angles in Alternate Segment

Definition of Alternate Segment

In a circle, when a tangent touches the circle at a point, it forms a segment with a chord drawn from that point. This segment is called the alternate segment. The alternate segment lies on the opposite side of the chord from the tangent.

More precisely:

• The tangent is a straight line that touches the circle at exactly one point.
• The chord is a line segment with both endpoints on the circle.
• The alternate segment is the region inside the circle that lies opposite the tangent, separated by the chord.

The angle formed between the tangent and the chord at the point of contact is related to an angle inside the alternate segment of the circle.

Alternate Segment Theorem

The Alternate Segment Theorem states:

The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment of the circle.

In other words, if a tangent touches the circle at point $A$, and $AB$ is a chord, then the angle between the tangent and chord at $A$ is equal to the angle subtended by chord $AB$ in the alternate segment of the circle.

This theorem applies only to angles inside the circle and helps find unknown angles by relating the tangent-chord angle to an angle elsewhere in the circle.

For example, if the angle between the tangent and chord at point $A$ is $40^\circ$, then the angle in the alternate segment subtended by the chord $AB$ is also $40^\circ$.

Identifying Angles in Problems

To use the Alternate Segment Theorem effectively, follow these steps:

• Identify the tangent: Look for the line that touches the circle at exactly one point.
• Identify the chord: Find the line segment from the tangent point to another point on the circle.
• Locate the angle between the tangent and chord: This is the angle formed outside the circle at the point of contact.
• Find the alternate segment angle: This is the angle inside the circle on the opposite side of the chord from the tangent.
• Apply the theorem: Set the angle between the tangent and chord equal to the angle in the alternate segment.

This approach helps in calculating unknown angles in circle problems involving tangents and chords.

Applications and Examples

The Alternate Segment Theorem is often combined with other circle theorems to solve complex angle problems. It is also used in proofs and reasoning about circle geometry.

For instance, if you know the angle between a tangent and chord, you can find the angle inside the circle without directly measuring it, which is especially useful in diagrams where direct measurement is difficult.

Here is a simple example: If the angle between the tangent and chord at the point of contact is $50^\circ$, then the angle in the alternate segment subtended by the chord is also $50^\circ$.

Below is a learning example demonstrating how to apply the theorem:

Example: A tangent touches a circle at point $A$. The chord $AB$ is drawn, and the angle between the tangent and chord at $A$ is $50^\circ$. Find the angle in the alternate segment subtended by chord $AB$.

Solution: By the Alternate Segment Theorem, the angle in the alternate segment is equal to the angle between the tangent and chord.

Therefore, the angle in the alternate segment is also $50^\circ$.