Lines and Angles | CBSE Class 6 Maths Notes

These notes cover points, line segments, lines, rays, naming angles, comparing turns, rotating arms, right and straight angles, degrees, protractors, angle bisectors, drawing and classifying angles, clock hands, estimation games and angle puzzles.

What are points and line segments?

How does a point show a location?

A point fixes a precise location. It has no length, breadth or height. A small pencil dot represents a point on paper, but the dot must be imagined as invisibly thin. The visible mark helps us locate the point.

The tip of a compass, the sharpened end of a pencil and the pointed end of a needle give models of a point. These objects help us imagine a location; their physical size does not give a mathematical point any dimensions.

Use a capital letter to name a point. For example, Z, P and T are point labels, read as Point Z, Point P and Point T. A label distinguishes one location from another without changing what the point represents.

What joins two points by the shortest route?

A line segment is the shortest path joining two points, including those points. If A and B name its two ends, the segment can be called segment AB or segment BA. Its end points are the locations where it begins and ends.

Definition: A line segment joins two end points by a straight path and includes both end points. Reversing the order of its two labels names the same segment.

A fold in a piece of paper gives a crease that models a line segment. When several routes connect A and B, distinguish the straight joining path from routes that bend or curve. The straight route is the shortest one.

Points A and B are connected by several drawn routes, including a straight route and bent or curved routes. The straight connection represents the line segment joining A and B.

See Fig. 2.1 in your NCERT textbook

How do lines, rays and line segments differ?

Property: Two points determine a unique line

A line continues endlessly in both directions. Imagine extending segment AB beyond A and beyond B without stopping. The result is line AB. The same line can also be named with a small letter, such as m.

Through two distinct points, there is one unique line. Here distinct means that the two points occupy different locations. A drawing shows part of the line because an endless line cannot be drawn completely on a sheet of paper.

One marked point does not determine a unique line: many lines can pass through that location in different directions. Adding a second distinct point fixes the line through both. This distinguishes the one-point question from the two-point question.

Where does a ray begin?

A ray begins at one point and continues endlessly in one direction. Its beginning is its starting point, also called its initial point. In ray AP, A is the starting point and P is another point along its path.

Light from a lighthouse, a torch and the Sun gives models for rays. The important geometrical idea is a starting location together with a direction of continuation. The first letter in a ray's name therefore matters.

Line m passes through A and B with arrowheads at both ends of the drawing. Ray AP begins at A, passes through P and has an arrowhead beyond P.

See Figs. 2.2 and 2.3 in your NCERT textbook

If ray OA starts at O and passes through B and A, it can also be called ray OB. Both names begin at O and follow the same direction. Ray AO starts at A, so reversing the letters does not name the original ray.

What is an angle and how should it be named?

Which parts form the angle?

An angle is formed by two rays with a common starting point. This shared point is the vertex. The rays are the arms. To recognise an angle, first locate the shared starting point, then follow the two rays from it.

In an angle formed by rays BD and BE, B is the vertex, while D and E are points on the two arms. The symbol ∠ means “angle”. The angle can be written as ∠DBE or ∠EBD.

In a three-letter angle name, the middle letter identifies the vertex. Reversing the outside letters keeps the vertex in the middle. Thus ∠DBE and ∠EBD can name the same indicated angle, whereas moving B to an end changes the vertex being named.

Two rays begin at B. The upper ray passes through D and the lower ray through E. B is labelled vertex, and each ray is labelled arm.

See Fig. 2.8 in your NCERT textbook

What does the size of an angle describe?

The size of an angle describes the amount of rotation, or turn, needed about the vertex to move one ray to the other. It describes how far the arm turns, rather than how far along the arm a point lies.

A small curve near the vertex indicates the angle being considered. This is especially useful when several rays meet at the same point. A single vertex letter may then be unclear, so use a three-letter name and mark the intended turn.

Opening a book cover, opening scissors and moving the arms of a compass illustrate turning. Identify the joining point and the two arms before comparing the opening. A greater turn produces a greater angle.

How can angles be compared without measuring them?

How does superimposition work?

Superimposition means placing one figure over another to compare them. For angles, make the vertices coincide, meaning they occupy the same position. Align one arm of each angle, then compare the positions of their other arms.

1. Trace one angle so that it can be moved over the other.
2. Place its vertex directly on the vertex of the second angle.
3. Align one arm of the traced angle with one arm of the other.
4. Compare the other arms and the turns indicated by the angle curves.

For the compared openings, the angle needing the greater turn is larger. If both pairs of arms overlap after the vertices coincide, the angles are equal: they require the same amount of rotation.

Property: Arm length does not determine angle size

Making the drawn arms longer or shorter does not change their directions. It therefore does not change the turn between them. Do not judge an angle by the lengths of its visible arms or the separation of their distant ends.

The equality symbol = means “is equal to”. When two angles match after superimposition, their measures can be connected with this symbol. Equality concerns the turn even when the visible portions of their arms have different lengths.

How can a transparent circle help?

Place the centre of a transparent circle on the first angle's vertex. Mark where its arms meet the circle's edge. Move the same circle to the second angle, again matching its centre to the vertex and one mark to an arm.

The remaining mark lets you compare the second arms. Keeping the same circle provides a common reference for the two openings. The method transfers an angle for comparison without assigning a numerical measure to it.

How do rotating arms and paper folds explain special angles?

What do rotating arms show?

A pair of rotating arms can be made by inserting two paper straws into the arms of a paper clip. Their joining point models a vertex. Turning the straws changes the opening and therefore changes the angle.

Compare several such models by superimposition. For the slit activity, trace one model on cardboard and cut an angle-shaped slit. Keep each model's angle fixed while checking which one matches the slit.

Note: A pair of arms passes through the slit when its angle equals the slit angle. This depends on the angle rather than arm length, as long as the arms are shorter than the length of the slit.

Property: A straight angle contains two right angles

A straight angle is a half turn. Its arms lie along a straight line in opposite directions. If the angle is named ∠AOB, O is its vertex and rays OA and OB are its arms.

To divide it equally, fold the paper so that arm OB overlaps arm OA. The crease through O divides the straight angle into two equal angles. Each is a right angle, so each represents a quarter of a full turn.

A right angle resembles an L, but its exact condition is that it equals half a straight angle. Its appearance may change when the paper is turned; the amount of rotation between its arms does not.

Perpendicular lines are lines that meet at right angles. Paper folding helps explain why the angles are equal: the matching parts overlap. The crease provides a practical way to create an exact right angle rather than estimate one by sight.

How are angles measured in degrees?

What is one degree?

A full turn is divided into 360 equal angular parts. Each part is one degree, written 1°. The symbol ° means degrees, the unit used here for measuring angles. An angle containing 30 such unit parts measures 30°.

A full turn returns the moving ray to its starting direction after one complete rotation. It measures 360°. A straight angle takes half of that rotation, while a right angle takes one quarter.

In calculations, ÷ means division into equal parts, × means multiplication, and − means subtraction. These operations help convert equal parts of a turn into degrees or find a remaining part of an angle.

Worked example 1. A full turn measures 360°. Find the measure of a straight angle, which is half a full turn.

Answer: Divide the full turn into two equal parts. 360° ÷ 2 = 180°. Therefore a straight angle measures 180°.

Worked example 2. Two equal right angles make a straight angle of 180°. Find the measure of one right angle.

Answer: 180° ÷ 2 = 90°. A right angle measures 90°, which is also one quarter of 360°.

These reference angles help you estimate other openings. Compare a turn with a quarter turn or half turn before measuring it. An estimate gives a rough expectation; a measurement gives the numerical angle using an instrument.

The reason why a full turn is divided into 360° is not fully known. Perhaps the most important practical reason for its continued use is that 360 divides evenly by every number up to 10 except 7.

How do you measure an angle with a protractor?

What do the markings mean?

A protractor is an instrument for measuring angles. A circular protractor divides a full turn into 360 equal parts. A semicircular protractor, shaped like half a circle, divides a straight angle into 180 equal parts.

A scale is an ordered set of degree markings. The unlabelled semicircular scale has long marks at intervals of 10° and medium marks halfway between them, at intervals of 5°. Smaller divisions allow single degrees to be counted. Labels make those divisions easier to read.

A labelled protractor has two scales. One increases from right to left, and the other increases from left to right. Choose the scale that starts at zero on the angle's aligned arm.

1. Find the angle's vertex and place the protractor's centre exactly there.
2. Turn the protractor until one arm passes through its zero-degree mark.
3. Follow the scale that begins at that zero, moving towards the other arm.
4. Read the degree marking where the other arm crosses the scale.

What if neither arm is at zero?

If the vertex is correctly centred, read both arms on the same scale and find the difference for the opening between them. Do not combine a reading from one scale with a reading from the other.

Worked example 3. Rays OT and OS start at vertex O. They cross the same outer protractor scale at 20° and 55°. Find the smaller angle ∠TOS.

Answer: The units between the readings give the angle. 55° − 20° = 35°. Thus ∠TOS measures 35°.

Before accepting a reading, check the centre, zero alignment and scale. Also compare the result with the opening you estimated. Correct placement and a consistent scale are necessary for the number to represent the intended turn.

How can you make a paper protractor and bisect an angle?

What does bisecting mean?

Bisecting an angle means dividing it into two equal angles. The line that does this is its angle bisector. Folding one arm onto the other gives a crease along this line because the two parts match.

To make a paper protractor, begin with a paper circle. Fold it into two equal halves and cut along the crease to obtain a semicircle, or half circle. This represents a half turn.

1. Write 0° at the semicircle's bottom right corner and 180° at its bottom left corner.
2. Fold the semicircle in half to obtain a quarter circle, then mark 90° at the top when opened.
3. Fold in half again to obtain creases corresponding to 45° and 135°.
4. Make another half fold, then open the sheet and mark the additional divisions.

Worked example 4. Repeated equal folds divide a straight angle of 180° into eight equal angles. Find each angle.

Answer: 180° ÷ 8 = 22.5°. Each neighbouring pair of fold directions therefore encloses an angle of 22.5°.

Figure 2.19 shows rays from centre O towards points A to I around a semicircle. Figure 2.20 marks 0°, 22.5°, 45°, 67.5°, 90°, 112.5°, 135°, 157.5° and 180°.

See Figs. 2.19 and 2.20 in your NCERT textbook

The folding process explains how equal angular divisions can be produced. Each new half fold divides an existing angle into equal parts. The 45° division is half of 90°, and the next half fold produces 22.5°.

A handmade protractor shows the angles supplied by its folds. A standard protractor supplies finer degree markings. Choose the instrument by considering whether its marked divisions let you read the angle you are trying to measure.

How do you draw an angle of a given measure?

How do you choose the vertex and base arm?

To draw an angle, decide its name and measure first. In ∠TIN, the point I is the vertex. T and N are points on its arms, so the arms are rays IT and IN. The middle letter fixes where the protractor's centre must go.

Choose ray IN as the base arm, meaning the reference arm from which the turn is measured. The second arm must start at I and follow the direction marked for the required number of degrees.

Worked example 5. Draw ∠TIN with measure 30°, using ray IN as the base and I as the vertex.

Answer: Draw ray IN. Centre the protractor at I, align IN with zero and mark T at 30° on that scale. Use a ruler to draw the second arm from I through T. The resulting angle is ∠TIN = 30°.

Which construction checks matter?

1. Check that the base begins at the intended vertex.
2. Place the centre of the protractor on that vertex, rather than elsewhere along the arm.
3. Read from the zero aligned with the base and mark the required degree value.
4. Join the vertex to the marked point and indicate the intended angle with a curve.

The new arm's direction determines the angle. Its drawn length does not set the angle's size. Extending the arm along the same straight path leaves the required turn unchanged.

The same method can draw angles such as 110°, 40°, 75°, 112° and 134°. Before drawing, compare the requested number with 90° and 180°. Afterwards, check that the opening has the expected size and that the vertex remains the middle letter in its name.

How are acute, obtuse and reflex angles classified?

Which boundaries define each type?

An acute angle measures more than 0° and less than 90°. Its turn is smaller than a right angle. An obtuse angle measures more than 90° and less than 180°, so its turn lies between a right angle and a straight angle.

A reflex angle measures more than 180° and less than 360°. It is larger than a straight angle but smaller than a full turn. A curve drawn around the vertex identifies this larger turn between the arms.

The boundaries matter. Exactly 90° is right, not acute or obtuse. Exactly 180° is straight, not obtuse or reflex. A complete turn of 360° is also outside the reflex range.

Does turning the page change the type?

The orientation of a drawing means the direction in which it is placed. Drawing an angle facing another way does not change its measure. Classify it by the indicated turn, not by whether its arms look upright or slanting.

The measures 40°, 50° and 75° are acute, while 110° and 130° are obtuse. These comparisons use the numerical boundaries directly. For a reflex drawing, check the marked curve carefully so that you do not accidentally measure the smaller opening.

How do clocks and games help you understand angle measures?

What can clock hands show?

Clock hands model two arms sharing the clock's centre as their vertex. At exact hours, the minute hand points to 12. The hour hand points to the hour number. The 12 equally spaced hour positions divide the full turn into equal parts.

Worked example 6. A clock has 12 equally spaced hour positions around 360°. Find the smaller angles between its hands at 1, 2, 4 and 6 o'clock.

Answer: Each hour gap measures 360° ÷ 12 = 30°. The required angles are 30°, 60°, 120° and 180° respectively, using one, two, four and six gaps.

How are estimation games scored?

In an estimation game, one team draws an angle with a protractor and the other guesses its size without measuring. Then the actual measure is shown. The score is the absolute difference: the non-negative gap between the guess and the measured value.

Worked example 7. An angle measures 49°, but the other team guesses 39°. Find its score when points equal the absolute difference in degrees.

Answer: 49 − 39 = 10. The team scores 10 points. A closer estimate would give a smaller gap and therefore a lower score.

In the second version, a team is told the required angle and draws it without a protractor. The other team measures the drawing. The score again records how far the drawing's measure differs from the target.

Worked example 8. The requested angle is 34°, and the drawn angle measures 25°. Calculate the score.

Answer: 34 − 25 = 9. The drawing is 9° away from the target, so the team scores 9 points.

Each team gets five turns, and the lower total score wins. Both versions connect visual judgement with measured turns. Checking an estimate afterwards helps develop a clearer sense of angle size.

How can known angles help solve angle puzzles?

How do you find the remaining part of an angle?

When a ray divides a known angle, use the given total and the known part to find the remaining part. First identify which turn has been divided. A straight angle supplies 180°, while a right angle supplies 90°.

Worked example 9. Points B, E and R lie in that order on a straight line. Ray ES forms a 90° angle with ray ER. Ray ET lies inside that right angle, and ∠TER = 80°. Find ∠BET and ∠SET.

Answer: Rays EB and ER form a straight angle, so ∠BET = 180° − 80° = 100°. Rays ES and ER form a right angle, so ∠SET = 90° − 80° = 10°.

The point E is the vertex of all these angles. Stating the straight line, the right angle and the position of ET supplies the information needed for both subtractions. The 80° value alone would not specify both remaining angles.

What can measuring triangles suggest?

A triangle is a closed figure with three straight sides. Measure its three inside angles with a protractor, write down the readings and add them. Repeat with other triangles before looking for a shared result.

A conjecture is a proposed general pattern suggested by observations. This activity asks you to make such a conjecture from measurements. Keep the observed readings separate from an explanation of why a result should hold in general.

What can equally spaced spokes tell you?

Worked example 10. The Ashoka Chakra has 24 equally spaced spokes around a full turn. Find the angle between neighbouring spokes and the largest acute angle between spokes.

Answer: Each gap measures 360° ÷ 24 = 15°. Five gaps give 5 × 15° = 75°. Six gaps give 90°, which is right rather than acute. The largest acute angle is therefore 75°.

An angle puzzle asks for an acute angle that remains acute when doubled, tripled and quadrupled, but becomes obtuse when multiplied by five. Here quadrupled means multiplied by four. The original measure must be greater than 18° and less than 22.5°.

The whole-number possibilities are 19°, 20°, 21° and 22°. Whole-number measures have no fractional part. Keep this condition explicit: if fractional measures are allowed, the entire range between 18° and 22.5°, excluding both ends, satisfies the puzzle.

Glossary

• Point — A precise location with no length, breadth or height, named using a capital letter.
• Line segment — The shortest straight path joining two end points and including both of those points.
• Line — A straight path that extends endlessly in both directions through its points.
• Ray — A portion of a line beginning at one point and continuing endlessly in one direction.
• Angle — A figure formed by two rays that have a common starting point.
• Vertex — The common starting point of the two rays forming an angle.
• Arms — The two rays starting from the vertex and forming the angle.
• Superimposition — Placing one figure over another to compare their positions, shapes or sizes.
• Right angle — An angle of 90°, equal to one quarter of a full turn.
• Straight angle — An angle of 180°, whose arms extend in opposite directions along a straight line.
• Acute angle — An angle measuring more than zero degrees and less than a right angle.
• Obtuse angle — An angle larger than a right angle but smaller than a straight angle.
• Reflex angle — An angle greater than 180° but less than a full turn of 360°.
• Protractor — An instrument with degree markings used to measure angles and draw specified angles.
• Angle bisector — A line that divides a given angle into two angles of equal measure.

Common errors and misconceptions

• Misconception: A point has the same width as its pencil dot. Correct: The dot represents a location; a mathematical point has no length, breadth or height.
• Misconception: Lines and line segments both stop at their named points. Correct: A segment has two end points, whereas a line continues endlessly beyond the labelled points in both directions.
• Misconception: Reversing a ray's letters leaves it unchanged. Correct: The first letter specifies its starting point. Ray OA begins at O, while ray AO begins at A.
• Misconception: Any letter can be the middle letter in an angle name. Correct: The vertex belongs in the middle. In ∠DBE, B is the vertex and BD and BE name the arms.
• Misconception: Longer drawn arms make a larger angle. Correct: The angle measures the turn between the arms. Extending them without changing their directions leaves the angle unchanged.
• Misconception: Either protractor scale gives the correct direct reading. Correct: Centre the protractor at the vertex and read the scale beginning at zero on the aligned arm.
• Misconception: An angle of exactly 90° is acute, and one of exactly 180° is obtuse. Correct: These are right and straight angles respectively; the acute and obtuse ranges exclude those boundaries.
• Misconception: The smaller opening between two rays is the angle to measure in every drawing. Correct: Follow the indicated curve. A reflex angle marks a turn greater than 180° and less than 360°.

Exam-style questions with model answers

Q1. State two differences between a line segment and a ray. [2 marks]

1. A line segment has two end points, while a ray has one starting point.
2. A segment stops at its ends, while a ray continues endlessly in one direction from its starting point.

Q2. Rays BD and BE begin at B, with D and E on the respective arms. Name the angle in two ways, identify its vertex and explain the position of the vertex letter. [3 marks]

1. The indicated angle may be named ∠DBE or ∠EBD. The outside letters identify points on its two arms, and their order can be reversed.
2. The vertex is B because both rays begin at B. Rays BD and BE are the arms forming the angle.
3. B must be the middle letter in either three-letter name. This position identifies the common starting point and distinguishes it from the points on the arms.

Q3. Explain how to compare two angles by superimposition and recognise when they are equal. [4 marks]

1. Trace one angle so it can be placed over the other. Keep its two arm directions unchanged while moving it.
2. Make the vertices coincide, then align one arm of each angle. This provides a common starting direction for comparing the turns.
3. Compare the other arms for the indicated openings. The angle requiring a greater turn from the aligned arm is larger.
4. If the vertices and both pairs of arms overlap, the angles are equal. The lengths of the drawn arms do not determine their sizes.

Q4. Describe how to draw ∠TIN = 30° with ray IN as the base, and give a final check of the drawing. [5 marks]

1. Draw ray IN with starting point I and a labelled point N along it. I will be the vertex of the required angle.
2. Place the centre of the protractor exactly on I. Align the zero-degree direction of the protractor with the base ray IN.
3. Use the scale that begins at zero along IN. Find its 30° marking and mark a point T in that direction.
4. Use a ruler to draw the second arm from I through T. The two arms of the required angle are now IT and IN.
5. Mark the intended opening and label it 30°. Check that the vertex is I, the middle letter in ∠TIN, and that the opening is smaller than a right angle.

Q5. Points B, E and R lie in that order on a straight line. Ray ES makes a 90° angle with ER. Ray ET lies inside ∠SER, and ∠TER = 80°. Find and classify ∠BET and ∠SET. [4 marks]

1. Rays EB and ER point in opposite directions along the given straight line. Therefore the angle ∠BER measures 180°.
2. The required angle ∠BET is the remaining part of that straight angle: 180° − 80° = 100°. It is obtuse.
3. The given right angle ∠SER measures 90°. Because ET lies inside it, ET divides it into the two angles ∠SET and ∠TER.
4. Subtract the known part to obtain ∠SET = 90° − 80° = 10°. This angle is acute because it lies between 0° and 90°.

Q6. A clock has 12 equally spaced hour positions around a full turn of 360°. At exact hours its minute hand points to 12 and its hour hand points to the hour number. Find one hour-gap angle and the smaller hand angles at 1, 2, 4 and 6 o'clock. [5 marks]

1. The 12 equal hour gaps divide a full turn. Each gap therefore measures 360° ÷ 12 = 30° at the centre of the clock.
2. At 1 o'clock, the hands point to 12 and 1. They are separated by one hour gap, giving an angle of 30°.
3. At 2 o'clock, the smaller opening spans two hour gaps. Its measure is 2 × 30° = 60°, which is an acute angle.
4. At 4 o'clock, the smaller opening spans four hour gaps. Its measure is 4 × 30° = 120°, which is an obtuse angle.
5. At 6 o'clock, the hands lie in opposite directions and span six hour gaps. The angle is 6 × 30° = 180°, a straight angle.

Q7. The Ashoka Chakra has 24 equally spaced spokes around 360°. Find the angle between neighbouring spokes and the largest acute angle between spokes, explaining the boundary. [3 marks]

1. Divide the full turn by the number of equal gaps: 360° ÷ 24 = 15°. Neighbouring spokes therefore enclose an angle of 15°.
2. An acute angle must be smaller than 90°. Five spoke gaps give 5 × 15° = 75°, which is acute.
3. Six gaps give 6 × 15° = 90°, a right angle. Thus five gaps give the largest acute angle, 75°, because another gap reaches the excluded boundary.

Q8. An estimation game awards points equal to the absolute difference in degrees between a guess and the measured angle. The angle is 49° and the guess is 39°. Calculate the score and explain what a smaller score means. [2 marks]

1. The absolute difference is 49 − 39 = 10, so the team scores 10 points.
2. A smaller score means a smaller gap between the estimate and the measurement, so the estimate was closer.

Key takeaways

• A point gives a precise location, and a line segment joins two end points by the shortest straight path.
• A line continues endlessly in both directions; a ray starts at one point and continues endlessly in one direction.
• An angle has two arms sharing a vertex, which is written in the middle of its three-letter name.
• Angle size depends on the turn between the arms, so changing their drawn lengths does not change the angle.
• A full turn measures 360°, a straight angle 180° and a right angle 90°.
• To measure directly, centre the protractor on the vertex and read from the zero aligned with one arm.
• An angle bisector divides an angle equally; repeated paper folds produce equal angular divisions for a handmade protractor.
• Classify the indicated angle using its degree measure, keeping the exact boundaries between acute, right, obtuse, straight and reflex angles.

Test yourself

What does a pencil dot represent in geometry?

It represents a precise location called a point, which has no length, breadth or height.

Ray OA begins at O and passes through B and A. Can it also be called ray OB?

Yes. Both names begin at O and describe the same direction along the ray.

Which letter identifies the vertex in ∠DBE?

B identifies the vertex because the vertex is the middle letter of the angle name.

What happens to an angle if its drawn arms are lengthened along the same directions?

Its measure stays unchanged because the turn between the two arms remains the same.

A straight angle is divided into two equal angles. What is each angle?

Each is a right angle of 90°, equal to a quarter of a full turn.

Two arms cross the same protractor scale at 20° and 55°. What is the smaller angle between them?

The smaller angle is 35°, found by subtracting 20° from 55° on that same scale.

What is an angle bisector?

It is a line that divides a given angle into two equal angles.

Why is an angle of exactly 180° not reflex?

It is a straight angle; a reflex angle must be greater than 180° and less than 360°.