Circle Theorems: Angles in the Same Segment

What Is the Theorem?

The angles in the same segment theorem tells us:

Angles formed on the circumference from the same chord, on the same side, are equal.

If you draw a chord across a circle and then draw two triangles using that chord and two different points on the same arc, the angles at the circumference will be equal.

This is one of the most visual theorems — and once you know what to look for, it’s hard to miss

Visual Summary

These angles are in the same segment, and they are always equal.

This is true no matter how big or small the arc is — as long as the angles are on the same side of the chord.

Example

Question:

In a circle, points $P$, $Q$, $A$, and $B$ lie on the circumference. Chord $PQ$ is drawn, and both points $A$ and $B$ lie above it on the same arc.

If: $\angle PAQ = 38\degree$

Find $\angle PBQ$.

Solution:

Since both angles are on the same arc from chord $PQ$, they are in the same segment.

$$\angle PBQ = 38\degree$$

Reason: Angles in the same segment are equal.

Worked Example

Question:

A circle has five points on its circumference: $A, B, C, D,$ and $E$.

You are given:

• $\angle AEB = 12\degree$
• $\angle BEC = 14\degree$
• $\angle CED = 73\degree$

Find: $\theta = \angle EBD$

Step 1: Understand What You’re Looking For

We need to find $\angle EBD$, which is formed from the ends of chord $ED$. We’re looking for another angle in the same segment — made from the same chord, on the same side.

We’re told: $\angle CED = 73\degree$ So, triangle $CED$ helps us out.

Let’s work within triangle $CED$. Since this triangle is inside a semicircle (noted from the previous diagram context), we might be able to find the third angle.

Let’s say we already know that angle $ECD = 17\degree$. (This would come from: $180\degree - 90\degree - 73\degree$)

Now notice:

$\angle EBD$ is also formed using the same chord $ED$

And it lies in the same segment as $\angle ECD$

So: $\theta = \angle ECD = 17\degree$

Reason: Angles in the same segment are equal.

Final Answer:

$$\boxed{\theta = 17^\circ}$$