Angles at the Centre and Circumference

1. Understanding the Key Concepts

Before we state the theorem, it's crucial to be precise about what "angle at the center" and "angle at the circumference" mean, and how they relate to the same "arc". An arc is just a part of the circumference of a circle. An angle is "subtended" by an arc if the ends of the arc are joined to the vertex of the angle.

The following video provides an excellent explanation of these concepts. It clearly distinguishes between angles subtended at the center and at the circumference, and how to identify the correct arc (major or minor) that forms them.

2. Defining the Theorem and its Variations

Now that we have the terminology clear, let's formally define the theorem and look at how it appears in diagrams.

To summarize the key points from the reading:

• The Theorem: The angle subtended by an arc at the center of a circle is twice the angle subtended by the same arc at any point on the remaining part of the circle.
• Formal Wording: For your exam, you must state: "The angle at the centre is twice the angle at the circumference."
• Key Patterns:
  1. The Arrowhead: This is the classic shape, where an acute or obtuse angle at the circumference is related to an angle at the center less than 180°.
  2. The Reflex Case: When the angle at the circumference is obtuse, it corresponds to the reflex angle at the center (the angle greater than 180°).

You might also recall our previous lesson on the angle in a semicircle. If the arc is a semicircle, the angle at the center is a straight line, which is 180°. According to our new theorem, the angle at the circumference should be half of that, which is 90°. This shows how the "angle in a semicircle" theorem is just a special case of this more general rule!

3. Proving the Theorem

A core part of extending your mathematical ability is understanding why theorems are true. The proof for this theorem is a classic piece of geometric reasoning that relies on properties of isosceles triangles—a concept you are already familiar with.

The following video walks through the standard proof. The strategy is to draw a line through the center and the point on the circumference, which splits the problem into two parts that are easier to solve.

The key steps in this proof are:

1. Construction: Draw a line from the angle at the circumference (C) through the center (O).
2. Isosceles Triangles: This creates two triangles. Both are isosceles because two of their sides are radii of the circle (e.g., OA = OC).
3. Exterior Angle Theorem: The exterior angle of a triangle is equal to the sum of the opposite interior angles. This allows you to relate the angle at the center to the angle at the circumference for each isosceles triangle.
4. Addition: By adding the results from the two triangles, we arrive at the final theorem: $\angle AOB=2\times \angle ACB$.

The worksheet below visually lays out these steps. It can be a helpful reference for how to structure such a proof on paper.

4. Applying the Theorem to Solve Problems

Now, let's put this theorem into practice. Problems will usually ask you to find a missing angle, and you will need to identify the "arrowhead" or "reflex diamond" pattern.

The next segment of the Radford Mathematics video demonstrates how to approach two typical problems.

Your Turn: Practice Problems

It is time to solve some problems yourself. The worksheet below contains a variety of questions that require you to apply the theorem.

Please attempt the following questions from the worksheet. For each one, calculate the missing angle and be prepared to state the reason for your answer.

1. Section A, Question 3 (a, b, c, d): These are four quick problems where you are given the angle at the center and need to find the angle at the circumference, or vice-versa.
2. Section B, Question 1: This question involves a reflex angle at the center.

Take a few minutes to work through them before checking the solutions below.

Solutions:

• 3a) $x=138^{\circ }÷2=69^{\circ }$.
  • Reason: The angle at the centre is twice the angle at the circumference.
• 3b) $x=45^{\circ }\times 2=90^{\circ }$.
  • Reason: The angle at the centre is twice the angle at the circumference.
• 3c) $x=127^{\circ }÷2=63.5^{\circ }$.
  • Reason: The angle at the centre is twice the angle at the circumference.
• 3d) $x=80.5^{\circ }\times 2=161^{\circ }$.
  • Reason: The angle at the centre is twice the angle at the circumference.
• Section B, 1) Angle ACB corresponds to the reflex angle at the center, which is $300^{\circ }$. Therefore, $ACB=300^{\circ }÷2=150^{\circ }$.
  • Reason: The angle at the centre is twice the angle at the circumference.

Conclusion

In this lesson, you have explored one of the most fundamental circle theorems. You have learned how to define it precisely, followed the logical steps of its proof, and applied it to find missing angles in various configurations.

Here are the key takeaways:

• The Theorem: An angle at the center of a circle is double the angle at the circumference if both are subtended by the same arc.
• Identification: Look for the characteristic "arrowhead" shape or the "diamond" shape which involves a reflex angle.
• Application: You can find the angle at the circumference by halving the angle at the center, or find the angle at the center by doubling the angle at the circumference.
• Formal Reason: In an exam, you must state, "The angle at the centre is twice the angle at the circumference."

In our next lesson, we will examine the relationship between angles that are in the same segment of a circle. You will find that the theorem we learned today provides a very straightforward way to prove the next one.