Alternate segment theorem

Objective. Connect tangent–chord angles with opposite arc angles.

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2. Worked example
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Learn: the key idea

The alternate segment theorem connects the angle between a tangent and a chord at the point of contact with the angle subtended by that chord in the alternate (opposite) segment of the circle. The tangent–chord angle on one side equals the inscribed angle in the segment on the other side of the chord. If the angle between the tangent and chord is 55°, the angle in the alternate segment subtended by that chord is also 55°.

Worked example

A tangent touches a circle at point T, and chord TB makes an angle of 48° with the tangent. Use the alternate segment theorem to find the angle subtended by TB at a point in the alternate segment.

1 $Identify the tangent-chord angle as 48°$
2 $Recall the alternate segment theorem: this equals the inscribed angle in the alternate segment$
3 $Locate the alternate segment on the opposite side of chord TB$
4 $State the inscribed angle there equals 48°$

Answer: 48°

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Common mistake. Students often match the tangent–chord angle with the angle in the same segment rather than the alternate one on the opposite side of the chord. Always identify the segment on the far side of the chord from the marked tangent angle.

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