Angles in the Same Segment

1. Defining the Theorem: Angles in the Same Segment

Let's begin by understanding what this theorem states. Imagine two points, A and B, on the circumference of a circle. They define an arc. Now, if you pick any two other points on the circumference, say C and D, the angles $\angle ACB$ and $\angle ADB$ will always be equal. These angles are said to be in the same "segment" of the circle.

The image below gives a perfect illustration of this property.

For your IGCSE exam, it's crucial to use the precise wording. The following resource from Save My Exams, tailored for the Cambridge syllabus, provides the exact definition and formal reason you'll need.

2. The Proof: A Direct Consequence

Why are these angles equal? The proof is a beautiful piece of logic that relies entirely on the theorem we covered in our last lesson: "the angle at the center is double the angle at the circumference."

The video below gives a concise and clear demonstration of this proof. As you watch, notice the simple but powerful strategy: both angles at the circumference are related back to the same angle at the center.

Let's quickly recap the steps of the proof:

1. Construct Radii: Introduce the center of the circle, $O$, and draw lines from $O$ to the endpoints of the arc (let's call them A and D). This forms the angle $\angle AOD$ at the center. Let's call this angle $x$.
2. Apply the Previous Theorem (First Angle): The angle at the circumference, $\angle ABD$, is subtended by the same arc as $\angle AOD$. Therefore, based on our last lesson, $\angle ABD=\frac{1}{2}x$.
3. Apply the Previous Theorem (Second Angle): The other angle at the circumference, $\angle ACD$, is also subtended by the same arc. Therefore, $\angle ACD=\frac{1}{2}x$.
4. Conclusion: Since both $\angle ABD$ and $\angle ACD$ are equal to $\frac{1}{2}x$, they must be equal to each other.

This proof is a great example of how mathematical principles build upon one another. You've used one theorem to prove another.

3. Application: Finding the "Bowtie"

Now for the practical part: using this theorem to solve problems. The key skill is pattern recognition. As you saw in the Save My Exams resource, this theorem often creates a shape resembling a bowtie or, as one instructor humorously calls it, "Spider-Man eyes."

The following video is excellent for training your eye to spot this pattern in different orientations, which is essential for IGCSE exam questions.

Your Turn: Practice Problems

It's time to put your knowledge to the test. The Siyavula online textbook has some well-designed practice questions. Try to solve the first two problems shown in the exercise set. Remember to state the reason for your answer.

Solutions Walkthrough:

• Problem 1: Angle $a$ and the $21^{\circ }$ angle are both subtended by the same arc. Therefore, $a=21^{\circ }$. The reason is: Angles in the same segment are equal.
• Problem 2:
  • To find $c$: Angle $c$ and the $24^{\circ }$ angle are subtended by the same arc. Therefore, $c=24^{\circ }$. Reason: Angles in the same segment are equal.
  • To find $d$: The $102^{\circ }$ angle is an exterior angle to the small triangle at the top. The exterior angle of a triangle equals the sum of the two opposite interior angles. So, $102^{\circ }=c+d$. Since we know $c=24^{\circ }$, we have $102^{\circ }=24^{\circ }+d$, which gives $d=78^{\circ }$.

This second problem is a great example of what you'll see in exams—a multi-step question that combines a circle theorem with other geometric rules.

Conclusion

Well done! You have now added another essential tool to your geometry toolkit. We have seen how this new theorem is not just a random rule to memorize, but a logical extension of the "angle at the center" theorem.

Here are your key takeaways from this lesson:

• The Theorem: Angles subtended by the same arc at the circumference are equal.
• The Formal Reason: For your IGCSE exam, you must state: "Angles in the same segment are equal."
• The Visual Cue: Look for the characteristic "bowtie" shape where four points on the circumference are connected.
• The Proof: The theorem is proven by showing that any two such angles are both equal to half of the same angle at the center.

In our next lesson, we will shift our focus slightly to another key component of circles: chords. We will explore the properties of isosceles triangles that are formed when two radii meet a chord, a concept that frequently appears alongside angle theorems in more complex problems.