# Angle in a semicircle is 90 degrees

Categories: gcse geometry

We can draw a triangle **ABC** where **AB** is a diameter of a circle, and **C** lies on the circumference of the circle:

The angle at the circumference **C** is a right angle. We say the *angle in a semicircle is a right angle*.

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## Proof

We can prove this as follows.

We start by drawing an extra line from the centre **O** to the point **C**.

Notice that the lines **OA**, **OB** and **OC** are all radii of the circle, and therefore all of equal length.

Looking at the triangle **AOC**, this is an isosceles triangle (from the rule 2 radii form an isosceles triangle). So the two angles at the circumference are equal (we will call them *a*):

By the same logic, the two angles at the circumference in triangle **BOC** are equal, and we will call them *b*:

Looking at the original triangle **ABC**:

The angle at **A** is *a*, the angle at **B** is *b*, and the angle at **C** is *a+b*. Since the three angles of a triangle add up to 180° we have:

Gathering the terms *a* and *b*:

Dividing both sides by 2 gives:

Since the angle at **C** is *a + b*, this proves that the angle at **C** is 90°.

## See also

- Parts of a circle
- Tangent and radius of a circle meet at 90°
- Two radii form an isosceles triangle
- Perpendicular bisector of a chord
- Angle at the centre of a circle is twice the angle at the circumference
- Angles in the same segment of a circle are equal
- Opposite angles in a cyclic quadrilateral add up to 180°
- Two tangents from a point have equal length

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