The Angle in a Semicircle is a Right Angle
A and B are the ends of a diameter, and C sits anywhere else on the circumference. The angle at C is always 90° — the triangle ACB is right-angled no matter where C goes.
Because the three angles of a triangle add to 180°, the two base angles at A and B must add to 90°. That is what makes this theorem useful: give a student one of them and they can work out the other.
How to Use
Interactive Exploration
Drag C around the circumference and watch the angle at C stay locked on 90° while the two base angles change. Drag A or B to spin the whole diameter — the right angle still holds.
Solving Problems
New positions sets a fresh question: one of the base angles is replaced by x while the other keeps its value. Subtract from 90° to find the missing angle, then click ∠BAC or ∠ABC to check.
Any of the three angles can be the unknown — including the right angle itself, where the question stops being arithmetic and becomes “why is C always 90°?”. The reveals track the diagram: drag a point while an answer is showing and the value updates with it.
The Style tab changes the arc and diameter colours and the background paper.
Teaching Notes
- Everything snaps to 2° steps, so all three angles are whole numbers and A + B = 90° exactly.
- C is held a little way clear of A and B so the triangle never collapses to a straight line.
- This is the special case of the angle at the centre theorem: the “arc” here is a semicircle, so the angle at the centre is a straight 180° and the angle at the circumference is half of it.