The Tangent-Chord Angle Theorem
Simple Explanation
Also called the Alternate Segment Theorem: the angle between a tangent line and a chord drawn from the point of tangency always equals the inscribed angle that same chord makes in the "alternate segment" β the arc on the opposite side of the chord.
Why Do We Need It?
This surprising equality connects an angle at the circle's edge (tangent-chord) to an angle deep inside the circle (inscribed) β a powerful tool for finding otherwise hard-to-reach angles.
See It
A circle with a tangent line at T, a chord TC, and the tangent-chord angle at T matching the inscribed angle at a point D on the far arc
Formula
The Tangent-Chord Angle Theorem
The angle between a tangent and a chord (drawn from the point of tangency) equals the inscribed angle in the alternate segment
Also called the Alternate Segment Theorem β the angle formed between a tangent line and a chord equals the angle that same chord subtends from any point on the arc on the OTHER side of the chord.
- tangent-chord angle
- β the angle between the tangent line and the chord, measured at the point of tangency
- alternate segment
- β the region of the circle on the opposite side of the chord from the tangent-chord angle being measured
When to use it: Whenever a tangent and a chord meet at a point of tangency, and you need to relate that angle to an inscribed angle elsewhere in the circle.
Worked Example
Apply the tangent-chord angle theorem
A tangent meets a chord at a point of tangency, forming a tangent-chord angle of 65Β°. Find the inscribed angle in the alternate segment.
Why Does This Work?
Stated simply: let the tangent-chord angle at T be ΞΈ, and draw the diameter from T. Since a tangent is perpendicular to the radius (and hence the diameter) at T, the angle between the chord and that diameter is 90Β°βΞΈ. The angle in the semicircle formed is 90Β° (angle in a semicircle), so the remaining angle of that right triangle is ΞΈ β and this angle turns out to be an inscribed angle subtending the same chord as the alternate segment's inscribed angle, so by the inscribed angle theorem, they must be equal.
Real-Life Example
Billiard ball rebound angles
A ball rolling along a curved cushion at the exact point where a straight cushion meets it tangentially reflects at a predictable angle related to the curve.
Understanding tangent-chord relationships helps predict and analyze such rebound angles in curved-boundary physics and engineering problems.
Practice
A tangent-chord angle measures 48Β°. Find the inscribed angle in the alternate segment.
HardCommon mistake
Matching the tangent-chord angle to the inscribed angle in the SAME segment (the near arc) instead of the alternate (opposite) segment β the equality specifically holds with the far, alternate segment.
Quick Review
- Tangent-chord angle = inscribed angle in the alternate (opposite) segment.
- Also called the Alternate Segment Theorem.
- Proof idea: uses the angle-in-a-semicircle theorem via the diameter from the point of tangency.