### 5) Tangents

There are two circle theorems involving **tangents**.

1. The angle between a tangent and a **radius** is 90°.

2. Tangents which meet at the same point are equal in length.

#### Example

Calculate the angles EFG and FOG.

Triangle GEF is an **isosceles** triangle. Angle FGE = angle EFG.

Angle FGE = angle EFG = (180 – 20) /2 = 80°

The angle between the tangent and the radius is 90°.

Angle EFO = angle EGO = 90°

The shape FOGE is a **quadrilateral**. The angles in a quadrilateral add up to 360°.

Angle FOG = 360 – 90 – 90 – 20 = 160°

#### Proof

The angle between the tangent and the radius is 90°.

Angle BCO = angle BAO = 90°

AO and OC are both radii of the circle.

Length AO = Length OC

Draw the line OB. It creates two triangles OCB and OAB. These share the length OB.

Triangles OCB and OAB are **congruent**. Two of the sides are the same length: OB = OB and OC = OA. One of the angles in each triangle is a right angle: OCB = OAB. Congruent triangles are identical.

So length CB = AB.