Naming Angles Using Three Letters
Question
A figure has rays meeting at different points. At vertex B, one ray goes towards A and another ray goes towards C. Write the correct name(s) for this angle. Also explain the rule for naming angles with three letters.
Solution — Step by Step
Step 1: Recall what an angle is.
An angle is formed when two rays start from the same point. That starting point is the vertex. The two rays are the arms of the angle.
Step 2: Identify the parts of this angle.
- Vertex: B (the point where the two rays meet)
- Arm 1: Ray BA (starting at B, going towards A)
- Arm 2: Ray BC (starting at B, going towards C)
Step 3: Apply the rule for naming angles.
The rule: the vertex is always the middle letter of the three-letter name.
The two letters on the outside can be any point on each arm (one letter on each arm). So the angle can be written as:
- ∠ABC (A is on one arm, B is the vertex in the middle, C is on the other arm)
- ∠CBA (same angle, just the outer letters swapped — both are correct)
Both ∠ABC and ∠CBA refer to the exact same angle.
Step 4: Can we use a shorter name?
If there is only one angle at vertex B (no other angles formed at B in the figure), we can also write it simply as ∠B. But if there are multiple angles at the same vertex, we must use the full three-letter name to avoid confusion.
∠ABC means: A → a point on one arm B → the VERTEX (always in the middle) C → a point on the other arm
∠ABC = ∠CBA (same angle, outer letters can be swapped)
Answer: The angle is ∠ABC (or equivalently ∠CBA). The vertex is B.
Why This Works
When we name an angle with three letters, the middle letter must be the vertex because the vertex is what both arms share — it is the “meeting point.” By placing the vertex in the middle, we clearly show: “This angle opens from the middle letter (vertex) towards the two outer letters (one on each arm).”
Think of it like directions: “Turn at the junction B, coming from A towards C.” The junction (B) is always in the middle of the description, not at the ends.
A memory trick: think of the vertex as the most important person — like the teacher standing in the middle. The two students (other points) stand on either side. The teacher (vertex) is always in the centre of the name: Student 1 — Teacher — Student 2, i.e., ∠ABC with B as the vertex.
Naming Angles in a Triangle
In triangle ABC, there are three angles. We name them as:
- ∠BAC or ∠CAB — angle at vertex A (A is in the middle)
- ∠ABC or ∠CBA — angle at vertex B (B is in the middle)
- ∠BCA or ∠ACB — angle at vertex C (C is in the middle)
In shorthand (when only one angle exists at each vertex), we write ∠A, ∠B, and ∠C.
Classifying the Angle
Just naming an angle is often not enough. We also classify it by its measure:
| Measure | Type |
|---|---|
| Less than 90° | Acute |
| Exactly 90° | Right angle |
| Between 90° and 180° | Obtuse |
| Exactly 180° | Straight angle |
In exam questions, you are often asked to: (1) name the angle using three letters, (2) name the vertex, (3) name the two arms. Practice all three parts together. For ∠ABC: the vertex is B, the arms are ray BA and ray BC.
Common Mistake
Mistake: Writing the vertex at the start or end of the three-letter name.
Wrong: ∠BAC (if the vertex is B — here B is at the start, which means A would be the vertex) Right: ∠ABC (B is in the middle, so B is the vertex)
Always place the vertex letter in the middle position of the three-letter angle name.