The angle at the junction

The angle between two tangents decides an angle at the centre.

The kite's four corners

Tangents from T touch a circle, centre O, at P and Q. T, P, O, Q are the corners of a kite. A kite has two pairs of equal sides next to each other. The tangents TP and TQ are one pair. The radii OP and OQ are the other.

Each tangent is perpendicular to the radius at the point of contact. So angle OPT = angle OQT = 90°. A quadrilateral's four angles add to 360°. Subtract the two right angles already counted:

angle PTQ + angle POQ = 360° − 90° − 90° = 180°.

The junction angle is angle PTQ, where the tangents meet. The centre angle is angle POQ. These two add to 180°. Two facts are used: the tangent–radius right angle and one angle sum.

Triangles OPT and OQT match by RHS. They share the hypotenuse OT, and OP = OQ. So OT cuts angle POQ in half. Then angle POT = half of angle POQ.

Worked pair

Let angle PTQ = 70°. Then angle POQ = 180° − 70° = 110°. Check the whole kite: 70 + 90 + 110 + 90 = 360. The four angles of a quadrilateral add to 360°.

Now take angle PTQ = 64°. Then angle POQ = 180° − 64° = 116°. Angle POT is half of that: 58°.

Two tangents make a kite. Tap the missing angle.

Try the lab to finish this lesson.