Notes For All Chapters – Maths Class 6 Ganita Prakash
📐 Ganita Prakash | Class 6 | Chapter 2
Lines and Angles
Explore points, line segments, lines, rays and angles — the building blocks of all geometry!
📍 Point
➖ Line Segment
↔️ Line
→ Ray
∠ Angles
📏 Protractor
⊥ Perpendicular
📋 Table of Contents
- 2.1 Point
- 2.2 Line Segment
- 2.3 Line
- 2.4 Ray
- Point vs Line Segment vs Line vs Ray — Quick Comparison
- 2.5 Angle — Vertex & Arms
- 2.6 Comparing Angles
- 2.8 Special Types of Angles
- 2.9 Measuring Angles & Protractor
- 2.10 Drawing Angles
- 2.11 All Types of Angles with Degree Ranges
- Chapter Summary
- Exam Practice Questions
2.1 Point
Mark a tiny dot on paper with a very sharp pencil tip. The sharper the tip, the thinner the dot. This gives us the idea of a point.
A point determines a precise location but has no length, no breadth, and no height.
🖊️ Naming a Point
Points are named using single capital letters. Examples: Point A, Point B, Point Z, Point P, Point T.
Remember
The dots on paper represent points, but they must be imagined to be invisibly thin. A real mathematical point has zero size.
The dots on paper represent points, but they must be imagined to be invisibly thin. A real mathematical point has zero size.
🌍 Real-Life Examples of a Point
Tip of a compassWhen you place a compass on paper, the sharp tip represents a point.
Sharpened pencil endThe very tip of a freshly sharpened pencil approximates a point.
Pointed end of a needleThe tip of a sewing needle models a point in space.
Star in the skyA distant star appears as a tiny dot — a point of light!
🔬
Zero Dimensions!
A point is 0-dimensional — it has no size at all. It only tells us where something is, not how big it is. Everything in geometry starts with a point!
A point is 0-dimensional — it has no size at all. It only tells us where something is, not how big it is. Everything in geometry starts with a point!
2.2 Line Segment
Fold a piece of paper and unfold it. The crease you see gives the idea of a line segment. A line segment has two end points.
A line segment is the shortest path between two points A and B. It is denoted by AB̄ or B̄A.
📌 Key Facts about Line Segments
- A line segment has two end points — let’s call them A and B.
- It is the shortest distance between those two points.
- You can measure its length using a ruler.
- It is written as AB or BA (both mean the same segment).
- Points A and B are called the end points of segment AB.
Shortest Path Trick
Between any two points A and B, there are infinitely many curved paths, but there is only one straight path — the line segment AB. This straight path is always the shortest!
Between any two points A and B, there are infinitely many curved paths, but there is only one straight path — the line segment AB. This straight path is always the shortest!
📐 Drawing a Line Segment
- Mark point A on your paper.
- Mark point B at some distance from A.
- Place a ruler connecting A and B.
- Draw a straight line between them. This is segment AB!
Bar Symbol
A bar over the two letters (Ā B̄) means line segment. The two letters at the ends = the two end points!
A bar over the two letters (Ā B̄) means line segment. The two letters at the ends = the two end points!
2.3 Line
Imagine taking a line segment AB and extending it endlessly in both directions — beyond A and beyond B, with no stopping point. This gives us a line.
A line extends infinitely in both directions. It is written as ↔AB or by a lowercase letter like l or m.
🔑 Key Properties of a Line
- A line has no end points — it goes on forever in both directions.
- We cannot draw a complete picture of a line — we can only show a part of it with arrows at both ends.
- Any two distinct points determine a unique line passing through both of them.
- A line is denoted by letters like l, m, or by two points: ↔AB.
Through 1 Point
Infinitely many lines can pass through a single point. (Think of all possible directions from one spot!)
Infinitely many lines can pass through a single point. (Think of all possible directions from one spot!)
Through 2 Points
Only one unique line can pass through two given points. (This is a fundamental fact of geometry!)
Only one unique line can pass through two given points. (This is a fundamental fact of geometry!)
Infinite in Both Directions!
A line is like a road that goes on forever — with no beginning and no end. When we draw a line on paper, the arrows at both ends remind us that it continues beyond what we can see.
A line is like a road that goes on forever — with no beginning and no end. When we draw a line on paper, the arrows at both ends remind us that it continues beyond what we can see.
2.4 Ray
A ray is a portion of a line that starts at one point (called the starting point or initial point) and goes on endlessly in one direction.
A ray has one starting point and extends infinitely in one direction. Ray from A through P is written as →AP.
🌟 Real-Life Examples of a Ray
Sun Ray
Starts at sun, goes endlessly
Torch Beam
Starts at torch, goes forward
Lighthouse Beam
Starts at source, goes far
📝 Naming a Ray
A ray is always named with the starting point first, then another point on the ray. So ray →AP starts at A and passes through P.
Order Matters for Rays!
Ray →AP and ray →PA are different! The first letter is ALWAYS the starting point. You cannot write →AP as →PA.
Ray →AP and ray →PA are different! The first letter is ALWAYS the starting point. You cannot write →AP as →PA.
Ray OA can also be called OB!
If ray OA passes through points A and B (with B farther out), you can also call the same ray OB — because both A and B are on the path of the ray starting at O.
If ray OA passes through points A and B (with B farther out), you can also call the same ray OB — because both A and B are on the path of the ray starting at O.
Quick Comparison: Point, Segment, Line & Ray
Here is a clear comparison of all the basic geometric ideas covered so far:
| Concept | Notation | End Points? | Direction | Can be Measured? | Real-Life Example |
|---|---|---|---|---|---|
| Point | A | None (it IS a location) | No direction | No (has zero size) | Tip of a needle |
| Line Segment | AB̄ | Two (A and B) | No extension | Yes (has definite length) | Edge of a ruler, crease |
| Line | ↔AB or l | None (infinite both ways) | Both directions (↔) | No (infinite length) | Edge of a straight road |
| Ray | →AP | One (starting point only) | One direction (→) | No (infinite in one way) | Sun rays, torch beam |
Memory Trick — 0, 2, 0, 1 End Points!
Point = 0 ends | Line Segment = 2 ends | Line = 0 ends | Ray = 1 end (start only)
Point = 0 ends | Line Segment = 2 ends | Line = 0 ends | Ray = 1 end (start only)
2.5 Angle — Vertex & Arms
An angle is formed by two rays having a common starting point.
An angle = two rays sharing one starting point. Symbol: ∠ (e.g., ∠ABC or ∠CBA)
📌 Parts of an Angle
Vertex
The common starting point of the two rays. Always the middle letter when naming the angle. Example: In ∠ABC, B is the vertex.
The common starting point of the two rays. Always the middle letter when naming the angle. Example: In ∠ABC, B is the vertex.
Arms
The two rays that form the angle. Example: In ∠ABC, ray BA and ray BC are the arms. They start at the vertex B.
The two rays that form the angle. Example: In ∠ABC, ray BA and ray BC are the arms. They start at the vertex B.
✍️ Naming an Angle
- Use the symbol ∠ followed by three letters.
- The vertex is ALWAYS the middle letter: ∠DBE means vertex at B.
- ∠DBE and ∠EBD are the same angle (order of arm-points doesn’t matter).
- Sometimes an angle can simply be called “Angle B” if there’s no confusion.
- A small curved arc at the vertex indicates which angle we mean.
Common Mistake!
∠APC cannot be called ∠P alone if there are multiple angles at P! You need all three letters when there’s more than one angle at a vertex. The vertex must always be the middle letter.
∠APC cannot be called ∠P alone if there are multiple angles at P! You need all three letters when there’s more than one angle at a vertex. The vertex must always be the middle letter.
🔄 Angle as Rotation
The size of an angle is the amount of rotation or turn needed about the vertex to move the first ray to the second ray.
Vidya’s Notebook — Understanding Angles!
Think of Vidya opening her notebook cover. The more she opens it, the bigger the angle! Case 1 (barely open) = small angle. Case 6 (fully open past flat) = very large angle. The more she rotates the cover, the bigger the angle gets.
Think of Vidya opening her notebook cover. The more she opens it, the bigger the angle! Case 1 (barely open) = small angle. Case 6 (fully open past flat) = very large angle. The more she rotates the cover, the bigger the angle gets.
🌍 Real-Life Angles
Compass / DividerThe two arms and the joining point (vertex) form an angle. Opening more = bigger angle.
ScissorsThe two blades are the arms; the pivot screw is the vertex. Opening scissors wider increases the angle.
Clock HandsAt 1 o’clock, the hands form a 30° angle. The center of the clock is the vertex!
Open DoorThe door itself and the doorframe are the arms; the hinge is the vertex.
Key Insight
The size of an angle depends ONLY on the amount of rotation — NOT on how long the arms are! A long-armed angle and a short-armed angle can have the same measure if they open by the same amount.
The size of an angle depends ONLY on the amount of rotation — NOT on how long the arms are! A long-armed angle and a short-armed angle can have the same measure if they open by the same amount.
2.6 Comparing Angles
How do we know which of two angles is bigger? There are two main methods:
📋 Method 1: Superimposition
Place one angle on top of the other so their vertices overlap. The angle whose arm sticks out beyond the other is the larger angle.
- Trace one angle on a transparent sheet.
- Place it over the other angle, matching the vertices.
- Align one arm of each angle on top of each other.
- Look at the other arms — whichever is “outside” belongs to the larger angle.
- If both arms overlap perfectly → the angles are equal!
Equal Angles by Superimposition
Two angles are equal in size when superimposed, all three features match: (1) common vertex, (2) first arm overlaps, (3) second arm also overlaps.
Two angles are equal in size when superimposed, all three features match: (1) common vertex, (2) first arm overlaps, (3) second arm also overlaps.
📋 Method 2: Using a Transparent Circle
Place a transparent circle with its centre on the vertex of an angle. Mark where the arms cross the circle’s edge (points A and B). Now place the same circle on the other angle — if the second angle’s arm falls inside the arc AB, the second angle is smaller; if it falls outside, the second angle is larger.
ARM LENGTH DOESN’T MATTER!
The slit experiment proves this: only the angle between arms matters, not how long the arms are. Two rotating arms pass through a slit only if their angle EQUALS the slit’s angle!
The slit experiment proves this: only the angle between arms matters, not how long the arms are. Two rotating arms pass through a slit only if their angle EQUALS the slit’s angle!
🔬 The Slit Experiment
Make rotating arms (two straws + paper clip). Cut a slit in the shape of one angle in cardboard. A pair of rotating arms passes through the slit only when its angle equals the slit’s angle — proving that angle size is independent of arm length!
2.8 Special Types of Angles
Two very important special angles arise from full and half turns:
Straight Angle
= 180°
Half a full turn. Arms lie on a straight line. Like a flat, open book.
Right Angle
= 90°
Quarter turn. Looks like letter L. Two perpendicular lines meet here.
🔗 Connection: Full Turn → Straight → Right
Full turn = 360°
Straight angle = ½ of full turn = 180°
Right angle = ½ of straight angle = 90°
Right angle = ¼ of full turn = 90°
Straight angle = ½ of full turn = 180°
Right angle = ½ of straight angle = 90°
Right angle = ¼ of full turn = 90°
How to Get a Right Angle by Folding
Draw a straight angle AOB on paper. Fold the paper so that OB overlaps exactly with OA. The crease formed is OC, which divides the straight angle into two equal right angles (∠AOC = ∠COB = 90°).
Draw a straight angle AOB on paper. Fold the paper so that OB overlaps exactly with OA. The crease formed is OC, which divides the straight angle into two equal right angles (∠AOC = ∠COB = 90°).
⊥ Perpendicular Lines
Two lines that meet at a right angle (90°) are called perpendicular lines. Written as: line AB ⊥ line CD.
Why shouldn’t you argue with a 90° angle?
Because it’s always right! 😄 (A right angle = 90°, and “right” also means correct!)
Because it’s always right! 😄 (A right angle = 90°, and “right” also means correct!)
🏷️ Classifying All Angles
Acute Angle
0° < angle < 90°
Less than a right angle. Sharp, pointy. “Acute” means sharp!
Right Angle
= exactly 90°
Exactly a quarter turn. The “L”-shaped angle. Perfect corner!
Obtuse Angle
90° < angle < 180°
More than right, less than straight. “Obtuse” means blunt or dull!
Straight Angle
= exactly 180°
A half turn. Forms a straight line. Like opening a book flat on a table.
Reflex Angle
180° < angle < 360°
More than a straight angle! Goes “the long way around” — more than a half turn.
Acute: 0° < θ < 90° | Right: θ = 90° | Obtuse: 90° < θ < 180° | Straight: θ = 180° | Reflex: 180° < θ < 360°
🔤
Why “Acute” and “Obtuse”?
Acute means sharp in Latin — and indeed, acute angles are pointy and sharp-looking. Obtuse means blunt or dull — and obtuse angles look wide and dull, not sharp!
Acute means sharp in Latin — and indeed, acute angles are pointy and sharp-looking. Obtuse means blunt or dull — and obtuse angles look wide and dull, not sharp!
2.9 Measuring Angles — The Protractor
To measure angles precisely, we use a tool called a protractor. Angles are measured in degrees (°).
🔢 Why 360 Degrees?
A Pinch of History!
The division of a full turn into 360° goes back thousands of years. The ancient Indian text Rigveda speaks of a wheel with 360 spokes! Ancient calendars of India, Persia, Babylonia, and Egypt were based on 360 days in a year. Babylonian mathematicians also counted in 60s (sexagesimal numbers). The number 360 is special — it can be divided evenly by 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, and 24!
The division of a full turn into 360° goes back thousands of years. The ancient Indian text Rigveda speaks of a wheel with 360 spokes! Ancient calendars of India, Persia, Babylonia, and Egypt were based on 360 days in a year. Babylonian mathematicians also counted in 60s (sexagesimal numbers). The number 360 is special — it can be divided evenly by 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, and 24!
1 full turn = 360° | 1 straight angle = 180° | 1 right angle = 90° | 1 degree = 1° (one unit part of a full turn ÷ 360)
📊 Key Angle Measures at a Glance
| Fraction of Full Turn | Degrees | What it Looks Like |
|---|---|---|
| Full turn (1/1) | 360° | Complete circle, back to start |
| Half turn (1/2) | 180° | Straight line — straight angle |
| Quarter turn (1/4) | 90° | Right angle (L-shape) |
| One-eighth (1/8) | 45° | Half of a right angle |
| One-third (1/3) | 120° | Interior angle of equilateral triangle |
| One-sixth (1/6) | 60° | Clock hand from 12 to 2 |
| One-twelfth (1/12) | 30° | One hour on a clock face |
🔧 How to Use a Protractor
- Place the centre point (midpoint of the flat side) of the protractor exactly on the vertex of the angle.
- Align the baseline (0° line) of the protractor along one arm of the angle.
- Read the number on the scale where the second arm of the angle crosses the protractor scale.
- Use the correct scale: if the angle opens to the right, use the right-to-left scale; if it opens to the left, use the left-to-right scale.
Common Protractor Mistakes to Avoid!
(1) Centre not on vertex — always align the centre hole. (2) Baseline not along one arm — rotate the protractor till it is. (3) Reading the wrong scale (inner vs outer) — for angles opening left, use the outer scale; for angles opening right, use the inner scale (or vice versa depending on orientation). (4) Not extending a short arm — extend with a ruler if needed!
(1) Centre not on vertex — always align the centre hole. (2) Baseline not along one arm — rotate the protractor till it is. (3) Reading the wrong scale (inner vs outer) — for angles opening left, use the outer scale; for angles opening right, use the inner scale (or vice versa depending on orientation). (4) Not extending a short arm — extend with a ruler if needed!
Unlabelled Protractor
Has only marks, no numbers. Count the small marks — each small mark = 1°. Long marks = 10°. Medium marks = 5°.
Has only marks, no numbers. Count the small marks — each small mark = 1°. Long marks = 10°. Medium marks = 5°.
Labelled Protractor
Has two sets of numbers (0–180) going in opposite directions. This helps measure angles regardless of which direction they open!
Has two sets of numbers (0–180) going in opposite directions. This helps measure angles regardless of which direction they open!
✂️ Making Your Own Protractor by Folding!
- Cut out a circle → Full turn = 360°
- Fold in half → Semicircle, straight angle = 180°
- Fold in half again → Quarter circle, right angle = 90°
- Fold in half again → 45° and 135° appear!
- One more half fold → 22.5°, 67.5°, 112.5°, 157.5° appear!
✂️ Angle Bisector
The process of getting half of a given angle is called bisecting the angle. The line (ray) that divides an angle into two equal halves is called the angle bisector.
If ray OC bisects ∠AOB, then ∠AOC = ∠COB = ½ × ∠AOB
2.10 Drawing Angles with a Protractor
To draw a specific angle (e.g., ∠TIN = 30°) using a protractor, follow these steps:
- Draw the base arm: Draw ray IN using a ruler. This is your reference arm.
- Place the protractor: Put the centre of the protractor exactly on point I (the vertex). Align the baseline along ray IN.
- Mark the angle: Starting from 0°, count up to 30° on the protractor scale. Make a small dot/mark at 30°.
- Draw the second arm: Use a ruler to join the vertex I to the dot you marked. Label this ray IT.
- Result: ∠TIN = 30° is your required angle! ✔
Pro Tip — Which Scale to Use?
If you draw the base arm going RIGHT, use the BOTTOM row of numbers (0 at the right). If base arm goes LEFT, use the TOP row (0 at the left). Always start counting from 0!
If you draw the base arm going RIGHT, use the BOTTOM row of numbers (0 at the right). If base arm goes LEFT, use the TOP row (0 at the left). Always start counting from 0!
📐 Angles on a Clock — Fun Application!
| Time | Angle Between Hands | Type of Angle |
|---|---|---|
| 12:00 | 0° | No angle (zero angle) |
| 1:00 | 30° | Acute (360° ÷ 12 = 30°) |
| 2:00 | 60° | Acute |
| 3:00 | 90° | Right Angle |
| 4:00 | 120° | Obtuse |
| 6:00 | 180° | Straight Angle |
Why 30° per hour?
A full clock face = 360°. There are 12 hours. So each hour = 360° ÷ 12 = 30°. At 1 o’clock, the hands are 1 hour apart → 30°. At 2 o’clock → 60°. At 4 o’clock → 120°. At 6 o’clock → 180° (straight angle)!
A full clock face = 360°. There are 12 hours. So each hour = 360° ÷ 12 = 30°. At 1 o’clock, the hands are 1 hour apart → 30°. At 2 o’clock → 60°. At 4 o’clock → 120°. At 6 o’clock → 180° (straight angle)!
🌀 Ashoka Chakra Angle
Ashoka Chakra has 24 spokes. Angle between two adjacent spokes = 360° ÷ 24 = 15°
2.11 All Types of Angles — Complete Reference
| Type | Degree Range | Description | Example |
|---|---|---|---|
| Zero Angle | = 0° | Both arms overlap — no opening at all | Closed compass |
| Acute Angle | 0° < θ < 90° | Less than a right angle; sharp and pointy | 30°, 45°, 60°, 75° |
| Right Angle | = exactly 90° | Quarter turn; looks like letter L; marked with a small square □ | Corner of a book, window corner |
| Obtuse Angle | 90° < θ < 180° | Greater than right but less than straight; blunt-looking | 110°, 130°, 150° |
| Straight Angle | = exactly 180° | Half turn; arms form a straight line | Flat open book on table |
| Reflex Angle | 180° < θ < 360° | More than half turn; goes the “long way” around | 210°, 270°, 330° |
| Complete Angle | = 360° | Full turn; comes back to start | Spinning top making one complete spin |
Quick Memory Chart
0° → Zero | 0°–90° → Acute | 90° → Right | 90°–180° → Obtuse | 180° → Straight | 180°–360° → Reflex | 360° → Complete
0° → Zero | 0°–90° → Acute | 90° → Right | 90°–180° → Obtuse | 180° → Straight | 180°–360° → Reflex | 360° → Complete
🧩 Puzzle: Finding Missing Angles
When a ray OC divides a straight angle ∠AOB (= 180°), the two parts always add up to 180°:
If ∠AOC + ∠COB = 180° (because together they form a straight angle)
Example from the book: If ∠TER = 80° and ∠REB is a straight angle (180°), then:
∠TER = 80°
∠REB = 180° (straight angle)
∠BET = 180° − 80° = 100° (obtuse angle)
∠SET = 90° (since ES ⊥ EB, from the figure)
∠REB = 180° (straight angle)
∠BET = 180° − 80° = 100° (obtuse angle)
∠SET = 90° (since ES ⊥ EB, from the figure)
Chapter Summary
📍 Point
Exact location, zero size. Named with a capital letter. Denoted by A, B, C…
Exact location, zero size. Named with a capital letter. Denoted by A, B, C…
➖ Line Segment
Shortest path between 2 points. Has two end points. Denoted AB̄. Can be measured.
Shortest path between 2 points. Has two end points. Denoted AB̄. Can be measured.
↔ Line
Extends infinitely in both directions. No endpoints. Denoted ↔AB or letter l, m.
Extends infinitely in both directions. No endpoints. Denoted ↔AB or letter l, m.
→ Ray
One starting point; extends infinitely in one direction. Denoted →AP. Order matters!
One starting point; extends infinitely in one direction. Denoted →AP. Order matters!
∠ Angle
Two rays from common point. Vertex (middle letter) + two arms. Size = amount of rotation.
Two rays from common point. Vertex (middle letter) + two arms. Size = amount of rotation.
📏 Measuring
1° = 1/360th of full turn. Full = 360°. Straight = 180°. Right = 90°. Use protractor to measure.
1° = 1/360th of full turn. Full = 360°. Straight = 180°. Right = 90°. Use protractor to measure.
✂️ Angle Bisector
Ray that divides angle into 2 equal parts. Created by paper folding.
Ray that divides angle into 2 equal parts. Created by paper folding.
⊥ Perpendicular
Two lines meeting at 90°. Right angle is formed. Symbol ⊥.
Two lines meeting at 90°. Right angle is formed. Symbol ⊥.
🔺 5 Angle Types
Acute (<90°), Right (=90°), Obtuse (90°–180°), Straight (=180°), Reflex (180°–360°).
Acute (<90°), Right (=90°), Obtuse (90°–180°), Straight (=180°), Reflex (180°–360°).
⚡ Key Formulas
Full turn = 360°
Straight angle = 180° = ½ of full turn
Right angle = 90° = ¼ of full turn = ½ of straight angle
Ashoka Chakra: angle between adjacent spokes = 360° ÷ 24 = 15°
Clock: angle per hour = 360° ÷ 12 = 30°
Exam Practice Questions
Q1. What is a point? Give three real-life examples of a point.
Ans: A point determines a precise location but has no length, breadth or height. Examples: (1) Tip of a needle, (2) Sharp tip of a compass, (3) Sharpened end of a pencil.
Q2. How is a line different from a line segment?
Ans: A line segment has two fixed end points and a definite (measurable) length. A line has no end points — it extends infinitely in both directions and cannot be measured. A line segment AB is part of line AB.
Q3. Why can ray AP not be written as ray PA?
Ans: In a ray, the first letter is always the starting point. Ray AP starts at A and goes through P in that direction. Ray PA would start at P and go through A — a completely different ray in the opposite direction!
Q4. Name the vertex and arms of angle ∠PQR.
Ans: Vertex = Q (the middle letter). Arms = ray QP and ray QR. The angle is formed by these two rays meeting at Q.
Q5. What is the degree measure of: (a) a full turn (b) a straight angle (c) a right angle (d) half of a right angle?
Ans: (a) Full turn = 360°. (b) Straight angle = 180°. (c) Right angle = 90°. (d) Half of a right angle = 45°.
Q6. Classify these angles: 35°, 90°, 125°, 180°, 270°, 60°.
Ans: 35° → Acute. 90° → Right. 125° → Obtuse. 180° → Straight. 270° → Reflex. 60° → Acute.
Q7. How many lines can pass through (a) one given point, and (b) two given points?
Ans: (a) Infinitely many lines can pass through a single point (in all possible directions). (b) Exactly ONE unique line can pass through two given points.
Q8. The Ashoka Chakra has 24 spokes. What is the angle between two adjacent spokes? Is it acute or obtuse?
Ans: Angle = 360° ÷ 24 = 15°. Since 15° is less than 90°, it is an acute angle.
Q9. ∠TER = 80° and ∠REB is a straight angle. Find ∠BET.
Ans: ∠REB = 180° (straight angle). ∠BET = ∠REB − ∠TER = 180° − 80° = 100°. This is an obtuse angle.
Q10. Write the steps to draw ∠ABC = 65° using a protractor.
Ans: Step 1 — Draw ray BC (base arm). Step 2 — Place protractor centre on B, align base with BC. Step 3 — Starting from 0°, mark a point at 65° on the scale. Step 4 — Join B to the marked point. Label it A. ∠ABC = 65° is ready!
Q11. Why does the size of an angle NOT change when we make the arms shorter or longer?
Ans: The size of an angle depends only on the amount of rotation (turn) from one arm to the other — not on how long the arms are. The rotating arms / slit experiment proves this: only the angle between arms matters, not their length.
Q12. What are perpendicular lines? Give an example.
Ans: Two lines that meet at a right angle (exactly 90°) are called perpendicular lines. Example: The edges of a book meet at right angles — the horizontal and vertical edges are perpendicular. Written as line AB ⊥ line CD.
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