Class 6 Maths Chapter 2:
Lines and Angles
Theory • Examples • Questions • Solutions
What are Lines and Angles?
Lines and Angles (NCERT Ganita Prakash Class 6, Chapter 2) introduces the foundational elements of geometry. A point marks an exact location with no dimensions. A line segment connects two fixed points with the shortest straight path. A line extends infinitely in both directions with no endpoints. A ray has one fixed starting point and extends infinitely in one direction. An angle is formed when two rays share a common vertex — its measure in degrees represents the amount of rotation between the arms. The five key types are: zero, acute (<90°), right (90°), obtuse (90°–180°), and straight (180°) angles. Angles are measured using a protractor and can be bisected by folding or construction.
What You Will Learn
Chapter Notes
Complete section-by-section notes for Lines and Angles (pp. 13–54):
Point
A point marks an exact, precise location in space. It has no length, width, or thickness — it is perfectly dimensionless.
Points are named with capital letters: Point A, Point B, Point P. On paper, a point is represented by a small dot.
- A point has 0 dimensions (zero length, zero width)
- Named by a single capital letter
- Two distinct points determine a unique line
- A dot on paper represents (but is not exactly) a point
Line Segment
A line segment is a straight path between two fixed endpoints. It has a definite, measurable length — the shortest distance between its two endpoints.
Line segment AB (written as AB with a bar) is bounded by Point A and Point B. It does not extend beyond these points.
- The edge of a ruler
- A pencil (approximate line segment)
- Side of a square drawn on paper
Line
A line is a straight path that extends infinitely in both directions — it has no endpoints and no fixed length. It is represented with arrows at both ends.
A line passing through Points A and B is called line AB (or line BA). Any two distinct points determine exactly one unique straight line.
- A line has no endpoints (extends both ways)
- A line segment has 2 endpoints (bounded)
- A line segment is part of a line
- Lines are not measurable in length
Ray
A ray has one fixed starting point (the initial point) and extends infinitely in one direction only. It has one endpoint but no other boundary.
Ray AB starts at Point A and passes through Point B, extending infinitely beyond B. Ray BA would be a completely different ray (starting at B, going through A).
- Sunlight from the sun (starts at sun, travels outward)
- A laser beam (starts at the source, extends one way)
- The arm of a clock (starts at center, extends outward)
Angle
An angle is formed when two rays share a common starting point (vertex). The two rays are called the arms of the angle. The angle measures the amount of rotation from one arm to the other around the vertex.
Angle AOB (written as ∠AOB or ∠BOA) has vertex O, with ray OA and ray OB as its two arms. The vertex is always the middle letter in the angle name.
- Three letters: arm point, vertex (middle), arm point
- ∠AOB = ∠BOA (same angle, different arm order)
- The vertex letter is ALWAYS in the middle
- Do not write ∠OAB or ∠OBA (wrong — vertex not middle)
Comparing Angles & Making Rotating Arms
Angles can be compared without measuring by overlapping: place one angle over another with vertices aligned and one arm matching. The angle whose free arm is further from the reference arm is larger.
A rotating arm tester (two cardboard strips on a pin) lets you test angles physically. Crucially: the length of the arms does NOT change the angle size — only the rotation (opening) between the arms matters.
Drag the slider to open the rotating arm and observe how angle changes:
Special Types of Angles
Both arms point in same direction; no rotation
Sharp angle; less than a right angle
Quarter turn; arms perfectly perpendicular
Wider than right angle; less than straight
Half turn; arms form a straight line
More than half turn; larger than straight angle
| Type | Range | Real-world Example |
|---|---|---|
| Zero | 0° | Closed book lying flat |
| Acute | 0° < θ < 90° | Open scissor blades (small opening) |
| Right | θ = 90° | Corner of a square/rectangle |
| Obtuse | 90° < θ < 180° | Open laptop screen past 90° |
| Straight | θ = 180° | Flat opened book / straight road |
| Reflex | 180° < θ < 360° | Clock hands at 8:00 (the large gap) |
Measuring Angles with a Protractor
A protractor is a semi-circular instrument for measuring angles in degrees. It has two scales (inner and outer, both from 0° to 180°) to allow measurement from either direction.
Place the protractor center exactly on the vertex of the angle.
Align the 0° baseline of the protractor with one arm of the angle.
Read the degree value where the second arm crosses the curved scale.
Choose the correct scale: if the angle opens to the right, use the right-side 0°; if to the left, use the left-side 0°.
Drawing Angles
To draw a specific angle (e.g., 75°) precisely using a ruler and protractor:
Draw a ray OA — this is the first arm of your angle. Mark the vertex O clearly.
Place the protractor center on vertex O and align the 0° line with ray OA.
Find the required angle (e.g., 75°) on the correct scale and mark a point B at that degree.
Remove the protractor and draw ray OB from vertex O through point B. The angle AOB = 75°.
Angle Bisector
An angle bisector is a ray that divides an angle exactly into two equal halves. If ray OB bisects ∠AOC, then ∠AOB = ∠BOC = half of ∠AOC.
- Draw any angle on a piece of paper with vertex O.
- Fold the paper so that one arm lies exactly on top of the other arm.
- Press firmly and crease the fold line.
- The crease is the angle bisector — it creates two perfectly equal angles.
If ∠AOC = 80°, its bisector OB creates ∠AOB = ∠BOC = 40°.
Angles in Real Life
Each hour gap = 30° (360° ÷ 12). At 3:00 = 90°, at 6:00 = 180°
A fully open door swings through ~90°–180°. Door hinges act as the vertex
The blades form an angle at the rivet. The wider they open, the larger the angle
Road ramps form acute angles with the ground — typically between 5° and 30°
Worked Examples
7 fully solved problems with detailed step-by-step reasoning:
Question: For angle ∠PQR, identify the vertex and the two arms.
- In angle notation ∠PQR, the middle letter is always the vertex.
- Vertex = Q (middle letter of ∠PQR).
- The two arms are rays starting from the vertex: ray QP and ray QR.
Question: Points O, A, B are given where O is the vertex. Is ∠AOB a valid name? What about ∠OAB?
- The vertex MUST be the middle letter in the angle name.
- ∠AOB: middle letter is O. O is the vertex. ✓ Valid name.
- ∠OAB: middle letter is A. But A is NOT the vertex — O is. ✗ Invalid.
Question: Riya draws ∠ABC = 40° with 3 cm arms. Rohan draws ∠PQR = 40° with 8 cm arms. Who has the larger angle?
- The measure of an angle depends ONLY on the rotation between arms, not arm length.
- Both angles measure exactly 40°.
- The arm length (3 cm vs 8 cm) is irrelevant to the angle's measure.
Question: Classify these angles: 35°, 90°, 135°, 180°, 245°.
- 35°: Between 0° and 90° → Acute angle.
- 90°: Exactly 90° → Right angle.
- 135°: Between 90° and 180° → Obtuse angle.
- 180°: Exactly 180° → Straight angle.
- 245°: Between 180° and 360° → Reflex angle.
Question: What is the angle between the hands of a clock at 4:00? What type of angle is it?
- A clock face has 360° divided into 12 equal hour gaps.
- Each hour gap = 360° ÷ 12 = 30°.
- At 4:00, minute hand is at 12, hour hand is at 4. Gap = 4 hours.
- Angle = 4 × 30° = 120°.
Question: Two adjacent angles on a straight line measure 65° and θ. Find θ.
- A straight line forms a straight angle of exactly 180°.
- The sum of adjacent angles on a straight line is 180°.
- Missing angle = 180° - 65° = 115°.
Question: Ray OP bisects ∠MON where ∠MON = 70°. What is the measure of ∠MOP?
- An angle bisector divides an angle into two equal halves.
- ∠MOP = ∠NOP = ∠MON ÷ 2.
- ∠MOP = 70° ÷ 2 = 35°.
Hands-on Activities
Practical classroom and home activities based on NCERT Ganita Prakash:
Goal: Build a physical tool to test and compare angles.
Materials: Two cardboard strips or paper straws, one brass paper fastener/pin.
Steps: Pierce one end of both strips together with the fastener so they can rotate freely around the pivot vertex.
What You Learn: Open the tester to fit against table corners (90°), open books, or scissors to see how angle opening changes without changing the strip lengths.
Goal: Construct a degree-measuring disk from scratch.
Materials: Circular piece of paper, pencil.
Steps: Fold circle in half (180° straight line). Fold in half again (90° right angle). Fold once more (45° angle).
What You Learn: Angles are fractions of a complete 360° turn: 1/2 turn = 180°, 1/4 turn = 90°, 1/8 turn = 45°.
Goal: Find the exact bisector of an arbitrary angle.
Materials: Paper with any angle drawn on it.
Steps: Fold along the vertex so ray 1 exactly overlaps ray 2. Press firmly to crease.
What You Learn: The crease line creates two symmetrical, identical angles, proving it is the angle bisector.
Goal: Identify times that create specific angle families.
Materials: Toy analog clock or drawn clock faces.
Steps: Set clock to 1:00 (30° acute), 3:00 (90° right), 5:00 (150° obtuse), 6:00 (180° straight).
What You Learn: Each 1-hour interval on an analog clock equals 30° of rotation (360° ÷ 12).
Challenge Questions
Put your geometric deduction to the test:
The Ashoka Chakra on the Indian National Flag has 24 equally spaced spokes. What is the angle between any two consecutive spokes?
At 3:30, the minute hand points directly at 6. The hour hand has moved halfway between 3 and 4. What is the angle between the hands?
Key Takeaways
Frequently Asked Questions about Lines and Angles
Concise, student-friendly answers for Class 6 examinations and quick review:
Lines & Angles in Computational Thinking
The concepts from this chapter connect directly to how computers model spatial data. Angular turns are foundational in robotics navigation (turning algorithms use precise degree commands), coordinate geometry(vectors have direction and magnitude), and computer graphics (rotation matrices rely on angle arithmetic). The angle bisector concept appears in computational geometry algorithms for mesh triangulation and pathfinding.
Explore Class 6 Computational Thinking & AIClass 6 Maths Chapters
Continue through the NCERT Ganita Prakash handbook covering patterns, number play, data handling, and more.