Chapter 2
Lines and Angles
Learn everything about Points, Lines, Rays, Line Segments, Angles, Types of Angles, Parallel Lines, Perpendicular Lines with beautiful illustrations, solved examples, quizzes, notes and activities.
Topics
18 Detailed Topics
MCQs
50 Practice Questions
Examples
Solved Examples
Certificate
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Introduction
Geometry begins with simple objects like points, lines, rays and line segments. These concepts help us understand shapes, buildings, bridges, roads, maps, engineering designs, architecture and many other things in daily life.
Learning Outcomes
Identify
Identify points, rays, lines and line segments.
Measure
Measure angles accurately using a protractor.
Draw
Draw different kinds of angles.
Apply
Apply geometry in daily life situations.
18
Topics Covered
50+
Practice Questions
25
MCQs
18 min
Estimated Reading
Chapter Objectives
- Understand the meaning of Point, Line, Ray and Line Segment.
- Differentiate between various geometrical figures.
- Measure and draw angles accurately.
- Recognize different types of angles.
- Understand parallel, perpendicular and intersecting lines.
- Apply geometrical concepts in daily life.
Quick Facts
Fact 1
Geometry is one of the oldest branches of Mathematics.
Fact 2
Architects use angles every day while designing buildings.
Fact 3
Engineers depend on geometry for roads and bridges.
Fact 4
Every map uses geometrical concepts.
Before You Start
- Basic shapes
- Numbers
- Measurement
- Use of a ruler
Materials Required
Scale
Pencil
Protractor
Eraser
Point
A point represents an exact position. It has no length, width or thickness. A point is represented using a capital letter such as A, B or C.
Point
A Point represents an exact position or location. It has no length, breadth or thickness. Points are named using capital letters like A, B, C.
Line
A Line is a straight path extending endlessly in both directions. It has no endpoints.
Line Segment
A Line Segment is a part of a line. It has two fixed endpoints and its length can be measured.
Ray
A ray starts from one fixed point and extends endlessly in one direction.
Comparison
| Figure | Endpoints | Length |
|---|---|---|
| Point | 0 | None |
| Line | None | Infinite |
| Ray | 1 | Infinite |
| Line Segment | 2 | Finite |
Real-Life Examples
Railway Track
Looks like parallel lines.
Arrow
Example of a ray.
Scale
Represents a line segment.
Road
Long straight roads resemble lines.
Key Points
- A point has no dimensions.
- A line extends endlessly in both directions.
- A line segment has two endpoints.
- A ray has one endpoint.
- A line segment can be measured.
- A line cannot be measured because it is infinite.
Activity
- Point
- Line
- Ray
- Line Segment
- Parallel Lines
- Intersecting Lines
Did You Know?
Quick Check
- What is a point?
- How many endpoints does a ray have?
- Can a line be measured?
- What is a line segment?
- Give one real-life example of a ray.
What is an Angle?
An Angle is formed when two rays start from the same point. The common point is called the Vertex. Angles are represented using the symbol ∠.
Parts of an Angle
Vertex
The common endpoint where the two rays meet.
First Arm
The first ray of the angle.
Second Arm
The second ray of the angle.
Interior
The space between the two arms.
Naming an Angle
An angle can be named in different ways.
| Diagram | Name |
|---|---|
| ∠ABC | Vertex at B |
| ∠PQR | Vertex at Q |
| ∠A | If only one angle exists |
Measuring Angles
Angles are measured using a Protractor. The standard unit of measuring an angle is Degree (°).
| Unit | Symbol |
|---|---|
| Degree | ° |
- 30°
- 45°
- 60°
- 90°
- 120°
- 180°
- 360°
How to Measure an Angle?
- Place the centre of the protractor on the vertex.
- Align the base line with one arm of the angle.
- Read the value where the second arm crosses the scale.
- The reading gives the measure of the angle.
Important Facts
Degree
Angles are measured in degrees.
Complete Turn
One complete turn measures 360°.
Half Turn
Half turn measures 180°.
Quarter Turn
Quarter turn measures 90°.
🌍 Angles Around Us
Door
Opening a door forms an angle.
Scissors
The blades make different angles.
Clock
Clock hands create angles.
Book
An open book forms an angle.
🧪 Activity
- An open notebook
- A pair of scissors
- A door
- A clock at 3:00
- A clock at 6:00
💡 Remember
✔ Common endpoint is called the Vertex.
✔ Angles are measured in Degrees (°).
✔ Protractor is used to measure angles.
Types of Angles
Angles are classified according to their measurements. There are six main types of angles.
1. Acute Angle
An Acute Angle measures more than 0° but less than 90°.
2. Right Angle
A Right Angle measures exactly 90°.
3. Obtuse Angle
An Obtuse Angle measures more than 90° but less than 180°.
4. Straight Angle
A Straight Angle measures exactly 180°.
5. Reflex Angle
A Reflex Angle measures more than 180° but less than 360°.
6. Complete Angle
A Complete Angle measures exactly 360°.
Comparison of Angles
| Type | Measure | Example |
|---|---|---|
| Acute | Less than 90° | 45° |
| Right | 90° | 90° |
| Obtuse | Between 90° and 180° | 120° |
| Straight | 180° | 180° |
| Reflex | Between 180° and 360° | 250° |
| Complete | 360° | 360° |
🌍 Real-Life Examples
Door
Opening a door creates different angles.
Clock
Clock hands show different types of angles.
Scissors
The blades form acute and obtuse angles.
Book
An open book forms different angles.
💡 Memory Trick
- Acute → Small (<90°)
- Right → Exactly 90°
- Obtuse → Bigger than 90°
- Straight → 180°
- Reflex → Bigger than 180°
- Complete → 360°
Solved Example
Example
Identify the type of angle measuring 135°.
⚡ Quick Check
- Is 70° an acute angle?
- What is the measure of a right angle?
- Which angle measures exactly 180°?
- What is a complete angle?
- Give one example of a reflex angle.
Complementary Angles
Two angles whose sum is exactly 90° are called Complementary Angles.
Examples
- 20° + 70° = 90°
- 35° + 55° = 90°
- 40° + 50° = 90°
Supplementary Angles
Two angles whose sum is exactly 180° are called Supplementary Angles.
Examples
- 100° + 80° = 180°
- 110° + 70° = 180°
- 150° + 30° = 180°
Adjacent Angles
Adjacent angles are two angles that:
- Have a common vertex
- Have one common arm
- Do not overlap
Vertically Opposite Angles
When two lines intersect, the opposite angles formed are called Vertically Opposite Angles. These angles are always equal.
| Angle | Equal To |
|---|---|
| ∠A | Opposite Angle |
| ∠B | Opposite Angle |
Intersecting Lines
Two lines that meet at one point are called Intersecting Lines.
Scissors
Blades intersect each other.
Road Crossing
Roads intersect at junctions.
Parallel Lines
Parallel lines are lines that never meet even when extended forever.
Examples
- Railway Tracks
- Notebook Lines
- Window Bars
Perpendicular Lines
When two lines intersect at exactly 90°, they are called Perpendicular Lines.
Transversal
A transversal is a line that cuts two or more lines at different points.
When a transversal cuts parallel lines, many pairs of angles are formed.
🕒 Clock Angles
| Time | Angle |
|---|---|
| 3:00 | 90° |
| 6:00 | 180° |
| 9:00 | 90° |
| 12:00 | 0° |
Solved Examples
Example 1
Question: Find the complement of 35°.
Answer: 55°
Example 2
Question: Find the supplement of 125°.
Answer: 55°
💡 Remember
✔ Supplementary Angles → Sum = 180°
✔ Adjacent Angles → Common Arm
✔ Vertically Opposite Angles → Equal
✔ Parallel Lines → Never Meet
✔ Perpendicular Lines → Meet at 90°
Practice Questions
Very Short Answer Questions
- What is a point?
- Define a line.
- What is a ray?
- What is a line segment?
- How many endpoints does a ray have?
- Which instrument is used to measure angles?
- What is the unit of angle?
- What is a right angle?
- What is an acute angle?
- What is an obtuse angle?
Short Answer Questions
- Differentiate between a line and a line segment.
- Explain the parts of an angle.
- Write two examples of parallel lines.
- Write two examples of perpendicular lines.
- What are complementary angles?
- What are supplementary angles?
- What are adjacent angles?
- What are vertically opposite angles?
- What is a transversal?
- Explain intersecting lines with examples.
Long Answer Questions
- Explain all six types of angles with diagrams.
- Differentiate between acute, right and obtuse angles.
- Explain the importance of angles in daily life.
- Describe different types of lines.
- Write five real-life examples of angles.
Chapter MCQs
1. A line has
2. Which angle measures exactly 90°?
3. Complementary angles add up to
4. Supplementary angles add up to
5. Parallel lines
6. A complete angle measures
7. Vertically opposite angles are
8. Which instrument measures angles?
9. A ray has
10. A straight angle measures
Chapter Summary
- Point represents an exact position.
- Line extends infinitely in both directions.
- Line segment has two endpoints.
- Ray has one endpoint.
- Angles are measured in degrees (°).
- There are six major types of angles.
- Complementary angles total 90°.
- Supplementary angles total 180°.
- Vertically opposite angles are equal.
- Parallel lines never meet.
- Perpendicular lines intersect at 90°.
- Transversal cuts two or more lines.