A tangent touches the circle at one point and is perpendicular to the radius there, so the angle between them is 90°. Two tangents from the same outside point are equal in length.
The perpendicular from the center to a chord cuts the chord in half. The alternate segment theorem says the angle between a tangent and a chord equals the angle in the opposite segment. With tangents TA and TB and ∠ATB = 36°, the quadrilateral OATB has two 90° angles, so ∠AOB = 360 − 90 − 90 − 36 = 144°.
Worked example
TA and TB are tangents and ∠ATB = 50°. Find ∠AOB.
- ∠OAT = ∠OBT = 90° (tangent meets radius).
- 360 − 90 − 90 − 50 = 130.
Answer:130°
Key words
- tangent
- a line touching the circle once
- chord
- a line joining two points on the circle
- alternate segment
- the segment on the other side of the chord
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Common Core-referenced: HSG-C.A.2, HSG-C.A.4