RS Aggarwal Solutions for Class 6 Chapter 11 Line Segment, Ray and Line

Exercise 11.1

Question 1. Give three examples of angles from your environment.
Answer: Three instances of angles in the world around us include the following:
(i) The angle created where the minute and hour hands meet on a clock face.
(ii) The angle formed at the corner where two walls of a room meet.
(iii) The angle created between two neighbouring fingers on your hand.
In simple words: Look around - a clock's hands make an angle, the corner of a room makes an angle, and your fingers make angles too.

Exam Tip: Use real-world, relatable examples that are easy to visualize and clearly show the angle being formed.

Question 2. Write the arms and the vertex of ∠LMP given in the figure.
Answer: The arms of ∠LMP are MP and ML. The vertex of the angle is the point M.
In simple words: The two sides of the angle are called arms, and the point where they meet is the vertex.

Exam Tip: Always identify the vertex first (the middle letter), then the two arms extend outward from it.

Question 3. How many angles are formed in the figures given? Name them. (fig. from book)
Answer:
(i) Three angles are formed, namely ∠ABC, ∠BAC, and ∠ACB.
(ii) Four angles are formed, namely ∠ABC, ∠ADC, ∠BCD, and ∠BAD.
(iii) Eight angles are formed, namely ∠ADC, ∠ACD, ∠DAC, ∠ACB, ∠ABC, ∠BAC, ∠BCD, and ∠BAD.
In simple words: Count every angle you can spot in each figure - include both small angles and larger ones made by combining rays.

Exam Tip: Systematically list all angles by examining each possible vertex and pair of rays; overlapping angles count separately.

Question 4. From figure, list the points which are: (fig. from book)
(i) in the interior of ∠P
(ii) in the exterior of ∠P
(iii) lie on ∠P
Answer:
(i) Points J and C are positioned inside ∠P.
(ii) Points D and B are positioned outside ∠P.
(iii) Points A, P and M are located on ∠P.
In simple words: Interior means inside the angle, exterior means outside, and on means touching the angle's rays or vertex.

Exam Tip: Draw or visualize the angle clearly to distinguish its interior region from the exterior region around it.

Question 5. In the figure, write another name for: (fig. from book)
(i) ∠1.
(ii) ∠2.
(iii) ∠3.
(iv) ∠4.
Answer:
(i) An alternate designation for ∠1 is ∠BOD.
(ii) An alternate designation for ∠2 is ∠BOC.
(iii) An alternate designation for ∠3 is ∠AOC.
(iv) An alternate designation for ∠4 is ∠AOD.
In simple words: The same angle can be named using any three points - the vertex in the middle and one point on each arm.

Exam Tip: Remember that the vertex letter must always be in the centre when naming an angle with three letters.

Question 6. In the figure, write another name for: (fig. from book)
(i) ∠1.
(ii) ∠2.
(iii) ∠3.
Answer:
(i) ∠BPE
(ii) ∠PQC
(iii) ∠DQF
In simple words: Each numbered angle can be renamed by selecting its vertex and any point from each of its two rays.

Exam Tip: Match the location of each numbered angle with the appropriate three-letter name by verifying the vertex position.

Question 7. In the given fig., which of the following statements are true: (fig. from book)
(i) Point B in the interior of ∠AOB
(ii) Point B in the interior of ∠AOC
(iii) Point A in the interior of ∠AOD
(iv) Point C in the exterior of ∠AOB
(v) Point D in the exterior of ∠AOC
Answer: Statements (ii), (iv), and (v) are correct.
Statements (i) and (iii) are incorrect since B is located on ∠AOB and A is located on ∠AOD.
In simple words: Check each point against each angle - is it inside, outside, or on the angle itself?

Exam Tip: For points on rays or vertices, they lie "on" the angle, not in its interior.

Question 8. Which of the following statements are true:
(i) The vertex of an angle lies in its interior.
(ii) The vertex of an angle lies in its exterior.
(iii) The vertex of an angle lies on it.
Answer: Statement (iii) is the only correct statement.
The vertex of an angle lies on the angle itself.
In simple words: The vertex is the point where the two rays meet - it sits right on the angle, not inside or outside it.

Exam Tip: Remember that a vertex is a boundary point of an angle, so it belongs to the angle itself, not to its interior or exterior.

Question 9. By simply looking at the pair of angles given in figure, state which of the angles in each of the pairs is greater. (fig. from book)
Answer:
(i) ∠AOB is greater than ∠DEF.
(ii) ∠PQR is greater than ∠LMN.
(iii) ∠UVW is greater than ∠XYZ.
In simple words: Compare the openness of each pair visually - the one that "opens wider" is the larger angle.

Exam Tip: Visual comparison requires looking at how far apart the rays are; the wider the opening, the larger the angle.

Question 10. By using tracing paper compare the angles in each of the pairs given in figure. (fig. from book)
Answer: When using tracing paper, we find that:
(i) ∠PQR is greater than ∠AOB.
(ii) ∠UVW is greater than ∠LMN.
(iii) ∠RST is greater than ∠XYZ.
(iv) ∠PQR is greater than ∠EFG.
In simple words: Trace one angle onto paper, then place it over the other to see which opening is wider.

Exam Tip: Tracing paper provides an exact overlay method for comparing angles - align the vertices and one ray to get an accurate result.

Exercise 11.2

Question 1. Give two examples each of right, acute and obtuse angles from your environment.
Answer: Two instances of a right angle in our surroundings are:
(i) The angle where two adjacent walls of a room meet forms a right angle.
(ii) The angle formed by two adjacent edges of a book creates a right angle.

Two instances of an acute angle in our surroundings are:
(i) The angle formed between two neighbouring fingers on your hand.
(ii) The angle between the two adjacent sides of the letter Z of the English alphabet.

Two instances of an obtuse angle in our surroundings are:
(i) The smaller angle created by two adjacent blades of a fan.
(ii) The smaller angle created by the two slanting sides of a roof of a hut represents an obtuse angle.
In simple words: Right angles are 90 degrees (like corners), acute angles are less than 90 degrees (sharp), and obtuse angles are between 90 and 180 degrees (wide).

Exam Tip: Use familiar, everyday objects to illustrate each angle type - this demonstrates clear understanding and is easy for examiners to verify.

Question 2. An angle is formed by two adjacent fingers. What kind of angle will it appear?
Answer: When two adjacent fingers meet, the angle will be an acute angle.
In simple words: Two fingers side by side make a sharp, narrow angle - that's an acute angle.

Exam Tip: Acute angles are visibly narrow and sharp; check that your answer reflects this characteristic.

Question 3. Shikha is rowing a boat due northeast. In which direction will she be rowing if she turns it through:
(i) a straight angle. (ii) a complete angle.
Answer:
(i) When Shikha rotates the boat through a straight angle or 180 degrees, she will be heading in the south-west direction.
(ii) When Shikha rotates the boat through a complete angle or 360 degrees, she will be moving in her original direction, which is the north-east direction.
In simple words: A 180-degree turn points you the opposite way; a 360-degree turn brings you back to where you started.

Exam Tip: Visualize the compass directions mentally or sketch them - a straight angle reverses direction completely, while a complete angle returns to the starting position.

Question 4. What is the measure of the angle in degrees between:
(i) North and West?
(ii) North and South?
(iii) North and South - East?
Answer: The angle's measurement is:
(i) The angle between North and West measures 90 degrees.
(ii) The angle between North and South measures 180 degrees.
(iii) The angle between North and South - East measures 135 degrees.
In simple words: North to West is a right angle (90°), North to South is a straight line (180°), and North to Southeast is in between (135°).

Exam Tip: Draw a compass rose and measure carefully - cardinal directions (N, S, E, W) form 90-degree angles with each other.

Question 5. A ship sailing in river Jhelam moves towards east. If it changes to north, through what angle does it turn?
Answer: If the ship is heading east and turns to face north, it rotates through a 90-degree angle.
In simple words: Going from east to north is a quarter turn, which is 90 degrees - a right angle.

Exam Tip: Use a compass diagram to visualize the turn - east to north always represents a 90-degree angle.

Question 6. You are standing in a class room facing north. In what direction are you facing after making a quarter turn?
Answer: After making a quarter turn or a turn of 90 degrees, I will be facing east if I turn to my right hand. Similarly, if I turn to my left hand, I will be facing west.
In simple words: A quarter turn from north gets you to either east (right) or west (left) depending on which way you rotate.

Exam Tip: Clarify the direction of rotation (right or left) to give a complete answer - a quarter turn is ambiguous without this detail.

Question 7. A bicycle wheel makes four and a half turns. Find the number of right angles through which it turns.
Answer: In a single rotation, a bicycle wheel covers 360°.

When we convert 360° into right angles, we get: 360°/90° = 4 right angles.

As a result, in four and a half rotations, the wheel will turn by (4 × 4.5) = 18 right angles.
In simple words: Each full turn is four right angles (90 degrees each). So 4.5 turns equals 4.5 times 4, which is 18 right angles.

Exam Tip: Convert the complete rotation to right angles first, then multiply by the number of turns - this ensures accuracy.

Question 8. Look at your watch face. Through how many right angles does the minute hand move between 8 O' clock and 10:30 O' clock?
Answer: The time period from 8:00 O'clock to 10:30 O'clock equals 2.5 hours, or two and a half hours.

In 1 hour, the minute hand rotates through a complete rotation, which is 360° or 360°/90° = 4 right angles.

Therefore, in 2.5 hours, the minute hand will rotate by 2.5 × 4 = 10 right angles.
In simple words: The minute hand makes one full turn per hour. In 2.5 hours, it makes 2.5 turns, which is 10 right angles.

Exam Tip: Calculate the time interval first, then convert hour count to right angles using the conversion 1 hour = 4 right angles.

Question 9. If a bicycle wheel has 48 spokes, then find the angle between a pair of adjacent spokes.
Answer: In a bicycle wheel, the total central angle is 360° and it contains 48 spokes.

Therefore, the angle between any two neighbouring spokes = 360/48 = 7.5°.
In simple words: Divide the complete circle (360°) by the number of spokes (48) to get the angle between adjacent ones.

Exam Tip: Always divide the total rotation equally among the number of divisions - this gives the angle between consecutive elements.

Question 10. Classify the following angles as acute, obtuse, straight, right, zero and complete angle:
(i) 118°
(ii) 29°
(iii) 145°
(iv) 165°
(v) 0°
(vi) 75°
(vii) 180°
(viii) 89.5°
(ix) 30°
(x) 90°
(xi) 179°
(xii) 360°
(xiii) 90.5°
Answer: An acute angle falls between 0° and 90°; an obtuse angle lies between 90° and 180°; a straight angle equals 180°; a right angle is 90°; a zero angle measures 0° and a complete angle is 360°.

(i) 118° is an obtuse angle.
(ii) 29° is an acute angle.
(iii) 145° is an obtuse angle.
(iv) 165° is an obtuse angle.
(v) 0° is a zero angle.
(vi) 75° is an acute angle.
(vii) 180° is a straight angle.
(viii) 89.5° is an acute angle.
(ix) 30° is an acute angle.
(x) 90° is a right angle.
(xi) 179° is an obtuse angle.
(xii) 360° is a complete angle.
(xiii) 90.5° is an obtuse angle.
In simple words: Sort each degree measurement into its category by comparing it against the boundary values - 0°, 90°, 180°, and 360°.

Exam Tip: Create a reference chart mentally: acute (0-90), right (90), obtuse (90-180), straight (180), and complete (360) to classify quickly and accurately.

Question 11. Using only a ruler, draw an acute angle, a right angle and an obtuse angle in your notebook and name them.
Answer: Three diagrams can be sketched:

— Acute angle ∠ABC: Draw two rays starting from point B, meeting at a point C and A such that the angle between them is less than 90°.
— Right angle ∠LMN: Draw two rays starting from point M, meeting at points L and N such that the angle between them is exactly 90° (mark with a small square).
— Obtuse angle ∠PQR: Draw two rays starting from Q, meeting at points P and R such that the angle between them is more than 90° but less than 180°.
In simple words: Use a ruler to draw two straight lines from a point - keep them close for acute, perpendicular for right, and wide but not opposite for obtuse.

Exam Tip: Label all vertices and rays clearly; for a right angle, include the small square symbol to show it's exactly 90°.

Question 12. State the kind of angle, in each case, formed between the following directions:
(i) East and West
(ii) East and North
(iii) North and North - East
(iv) North and South - East
Answer:
(i) East and west directions form an angle of 180°, which is a straight angle.
(ii) East and north directions form an angle of 90°, which is a right angle.
(iii) North and north-east directions form an angle of 45°, which is an acute angle.
(iv) North and south-east directions form an angle of 135°, which is an obtuse angle.
In simple words: Use compass knowledge - opposite directions give 180°, perpendicular directions give 90°, and diagonal directions give 45° or 135°.

Exam Tip: Commit to memory the key compass angle relationships: cardinal to cardinal = 90°, cardinal to opposite = 180°, cardinal to inter-cardinal = 45° or 135°.

Question 13. State the kind of each of the following angles:
Answer: Based on the given angle measures:
(i) An angle in the 0° to 90° range is called an acute angle.
(ii) An angle in the 90° to 180° range is called an obtuse angle.
(iii) An angle that equals 180° is called a straight angle.
(iv) An angle that equals 90° is called a right angle.
(v) An angle that equals 360° is called a complete angle.
In simple words: Different angles have different names based on how many degrees they measure. Small angles are acute, medium ones are obtuse, and larger ones have their own special names too.

Exam Tip: Memorise the degree ranges for each angle type - this is fundamental and appears frequently in geometry questions.

Question 1. The vertex of an angle lies
(a) in its interior
(b) in its exterior
(c) on the angle
(d) inside the angle
Answer: (c) on the angle
In simple words: The vertex is the point where two rays meet to form an angle. This point sits exactly on the angle itself, not inside or outside it.

Exam Tip: The vertex is always at the corner or meeting point of the two rays - this is a definition question, so be precise.

Question 2. The figure formed by two rays with the same initial point is known as
(a) a ray
(b) a line
(c) an angle
(d) a line segment
Answer: (c) an angle
In simple words: When two rays start from the same point, they create an angle. The shared starting point is the vertex.

Exam Tip: Remember that an angle requires two rays sharing a common starting point - without this, you don't have an angle.

Question 3. An angle of measure 0° is called
(a) a complete angle
(b) a right angle
(c) a straight angle
(d) none of these
Answer: (d) none of these
In simple words: A 0° angle doesn't fit the standard angle categories. It's actually called a zero angle.

Exam Tip: Zero angle is a special case - the two rays overlap completely, so no angle opening exists at all.

Question 4. An angle of measure 90° is called
(a) a complete angle
(b) a right angle
(c) a straight angle
(d) a reflex angle
Answer: (b) a right angle
In simple words: When two rays are perpendicular and form a 90° angle, this specific angle is known as a right angle.

Exam Tip: Right angle is one of the most commonly tested angle types - always verify that the measure is exactly 90°.

Question 5. An angle of measure 180° is called
(a) a zero angle
(b) a right angle
(c) a straight angle
(d) a reflex angle
Answer: (c) a straight angle
In simple words: When the two rays form a straight line, the angle between them is 180°, and this is called a straight angle.

Exam Tip: A straight angle looks like a straight line - the two rays point in completely opposite directions.

Question 6. An angle of measure 360° is called
(a) a zero angle
(b) a straight angle
(c) a reflex angle
(d) a complete angle
Answer: (d) a complete angle
In simple words: When a ray rotates all the way around and returns to its starting position, it has turned 360°, forming a complete angle.

Exam Tip: A complete angle is a full rotation - think of a clock hand going all the way around.

Question 7. An angle of measure 240° is
(a) an acute angle
(b) an obtuse angle
(c) a straight angle
(d) a complete angle
Answer: None of the given options are correct
In simple words: A 240° angle is larger than 180° but smaller than 360°, so it falls into the reflex angle category, which is not listed in the choices provided.

Exam Tip: When an angle exceeds 180° but stays below 360°, it's always a reflex angle - remember this distinction when options don't seem to fit.

Question 8. A reflex angle measures
(a) more than 90° but less than 180°
(b) more than 180° but less than 270°
(c) more than 180° but less than 360°
(d) none of these
Answer: (c) more than 180° but less than 360°
In simple words: A reflex angle is any angle that is bigger than a straight angle (180°) but has not made a complete rotation (360°).

Exam Tip: Reflex angles are the "large" angles - always remember they sit between 180° and 360°, never outside this range.

Question 9. The number of degrees in 2 right angles is
(a) 90°
(b) 180°
(c) 270°
(d) 360°
Answer: (b) 180°
In simple words: Since one right angle equals 90°, two right angles together equal 90° multiplied by 2, which gives 180°.

Exam Tip: Always multiply the degree value of one unit by the number of units needed - a quick calculation prevents careless errors.

Question 10. The number of degrees in 3/2 right angles is
(a) 180°
(b) 360°
(c) 270°
(d) 90°
Answer: The correct answer is 135°
In simple words: One right angle equals 90°. When we multiply 90° by 3/2 (which is 1.5), we get 135°.

Exam Tip: For fractional multiples of angle units, convert the fraction to a decimal or multiply carefully - this prevents computational mistakes.

Question 11. If bicycle wheel has 36 spokes, then the angle between a pair of adjacent spokes is
(a) 10°
(b) 15°
(c) 20°
(d) 12°
Answer: (a) 10°
In simple words: A complete wheel covers 360°. When the spokes divide it into 36 equal parts, each gap between neighbouring spokes measures 360° divided by 36, which equals 10°.

Exam Tip: In division problems involving equal parts around a circle, always divide the total degrees (360°) by the number of parts to find each individual angle.