Angles Theorems

What are Angles Theorems?

We are familiar with the parts of a circle, namely radius, diameter, chords and centre. Let’s discuss circle angle theorem. What are the other properties of a circle? What is circle theorem?

Circle theorems is finding the angle or a figure within a circle. Let’s recall the basic parts of a circle. The radius is a segment from the centre to any point around a circle. Diameter is a line segment within a circle, which passes through the centre. Diameter is equal to two radii. Chord is a is a line segment inside the circle that does not pass through the centre.

1. The angle in a semicircle is 90 degrees.

A semicircle is half of a circle. This theorem states that any angle formed at the same two points of a semicircle and with the centre point lying in the circumference is a right angle.

Arc ABC is a semicircle and forms a triangle ABC. Segment AC is a diameter. Therefore, ∠B is a right angle, which measures 90 degrees.

2. The angle at the centre is twice the angle at the circumference.

Angles from the same arc, an angle in which the vertex is in the centre, is twice the angle at the circumference.

∠A and ∠B are in the same arc, arc CD. ∠A vertex lies on the centre and ∠B vertex is in the circumference. This states that angle A measure twice of angle B (∠A = 2∠B).

Let’s look at an example: If ∠A = 110, what is the measure of ∠B?

∠A = 2∠B
110 = 2∠B                                                 Divide both sides by 2.
∠B = 55 degrees

Another example: If ∠B = 30 degrees, what is the measure of ∠A?

∠A = 2∠B
∠A = 2(30)
∠A = 60

3. Angles on the same arc are congruent

Any angles that have the same arc at the circumference are equal. The difference to the second rule, is that it has an angle in which the vertex is at the centre, while in the third rule all angle vertices lie on the circumference of the circle. 

Since arc AB is a common arc of ∠D and ∠E, then the angles are equal.

If ∠D = 30, then ∠E = 30.