📘 Chapter 2 – Lines and Angles
Textbook Page 13 Explanation (Part 1)
This page explains Angle, its vertex, arms (sides), how to name an angle, and how to identify the parts of different angles.
📐 What is an Angle?
"An angle is formed by two line segments that have a common point or rays that have a common starting point."
An angle is formed when two line segments or two rays meet at one common point.
When these two sides separate from each other, a space is formed between them. This space is called an angle.
- 🚪 Opening a door forms an angle.
- ✂️ Opening a pair of scissors forms an angle.
- 📖 Opening a book forms an angle.
- 🧭 Clock hands also form angles.
📏 Arms (Sides) of an Angle
"These line segments or rays that form the angle are called the arms of the angle or sides of the angle."
The two rays or two line segments that form an angle are called its arms or sides.
Every angle always has two arms.
- An angle has two arms.
- Without two arms, an angle cannot be formed.
📍 Vertex of an Angle
"The common point is called the vertex of the angle."
The point where both arms meet is called the vertex.
Every angle has only one vertex.
- Ray OA → First arm
- Ray OB → Second arm
- Point O → Vertex
✍️ Naming an Angle
"The name of the adjacent angle is read as angle AOB or angle BOA or angle O. Symbol (∠) is used to write the same as ∠AOB or ∠BOA or ∠O."
An angle can be named in different ways.
If there is only one angle at the vertex, we may simply write the name of the vertex.
In the textbook figure, the angle can be written as:
- ∠AOB
- ∠BOA
- ∠O
The symbol ∠ is used to represent an angle.
- The middle letter is always the vertex.
- ∠AOB → Vertex = O
- ∠BOA → Vertex = O
📝 Complete the Table
The textbook gives four angle figures. For each figure, we have to identify:
- ✔ Name of the angle
- ✔ Vertex
- ✔ Arms (Sides)
| Figure | Name of Angle | Vertex | Arms / Sides |
|---|---|---|---|
| (i) | ∠KYZ or ∠ZYK | Y | Ray YK, Ray YZ |
| (ii) | ∠RAD or ∠DAR | A | Ray AR, Ray AD |
| (iii) | ∠TAP or ∠PAT | A | Ray AT, Ray AP |
| (iv) | ∠MAP or ∠PAM | A | Ray AM, Ray AP |
🌟 Part 1 Summary
- An angle is formed by two rays or two line segments having one common point.
- The two rays are called the arms or sides of the angle.
- The common point is called the vertex.
- The symbol ∠ represents an angle.
- When writing an angle with three letters, the vertex is always written in the middle.
- Every angle has one vertex and two arms.
📘 Chapter 2 – Lines and Angles
Textbook Page 13 Explanation (Part 2)
This part explains why an angle should not always be named using only one letter, how to identify different angles in one figure, and how to compare angles.
💬 Understanding the Discussion Box
"In the adjacent figure can we write the name of the angle as ∠B? How do we know exactly which angle is ∠B? Because there are three rays with initial point B."
Look carefully at the figure.
Three rays start from the same point B.
- Ray BA
- Ray BC
- Ray BD
Since three rays start from B, more than one angle is formed.
If we simply write ∠B, nobody can understand which angle we are talking about.
📐 How Many Angles Are Formed?
"How many angles are there? Write the names of those angles."
There are three different angles in the figure.
| Angle | Formed Between |
|---|---|
| ∠ABC | Ray BA and Ray BC |
| ∠CBD | Ray BC and Ray BD |
| ∠ABD | Ray BA and Ray BD |
- As ∠ABC, ∠CBD and ∠ABD.
✍️ Why Are Three Letters Used?
"To represent a particular angle three letters are used to write an angle. Among them, the name of the vertex is in the middle."
Three letters help us identify the exact angle.
The middle letter always shows the vertex.
- ∠ABC → Vertex is B
- ∠CBD → Vertex is B
- ∠ABD → Vertex is B
📏 Comparing Angles
- Which angle is larger among ∠ABC and ∠ABD?
- Which angle is larger among ∠CBD and ∠ABD?
- How will you compare these angles?
✅ Question 1
Compare ∠ABC and ∠ABD.
The angle ∠ABD is larger because it opens wider than ∠ABC.
✅ Question 2
Compare ∠CBD and ∠ABD.
Again, ∠ABD is larger because it covers a bigger opening.
✅ Question 3
Angles are compared by observing how much they open.
The wider the opening between the two arms, the larger the angle.
- Small opening → Small angle.
- Large opening → Large angle.
❌ Common Mistakes
- ❌ Writing only ∠B when more than one angle is present.
- ❌ Writing the vertex as the first or last letter.
- ❌ Forgetting that the vertex must always be the middle letter.
- ❌ Thinking that longer arms mean a bigger angle. The size of an angle depends only on the opening, not on the length of its arms.
🌟 Page Summary
- When many angles share the same vertex, one letter is not enough to name an angle.
- Use three letters to identify the correct angle.
- The middle letter is always the vertex.
- The figure contains three angles: ∠ABC, ∠CBD and ∠ABD.
- ∠ABD is larger than both ∠ABC and ∠CBD.
- Angles are compared by looking at how wide they open.