CBSE Class 6 Mathematics Lines and Angles Assignment Set 01

Short Answer Questions

Question 1. Answer the following about a point.
(i) What does a point tell us, and why do we say that a point has no length, breadth or height?
(ii) Write two things from your pencil box that give you the idea of a point, and explain how we name points on paper.
Answer:
(i) A point tells us an exact location. It is only a mark of position, so it has no length, breadth or height. However sharp the pencil, a true point must be imagined as invisibly thin.
(ii) The sharpened end of a pencil and the pointed tip of a compass. A point is named by a single capital letter, such as point P or point T.
In simple words: A point is just a position with no size, and we name it with one capital letter.

Teacher's Note:
1. Our dot is only a picture of a point.
2. Capital letters for points, small letters for lines.

Question 2. Differentiate between a line segment and a line. Write any two differences and also show how each of them is denoted when the two points on it are S and T.
Answer:
(1) A line segment has two end points; a line has no end point.
(2) A line segment has a definite length that can be measured; a line goes on endlessly in both directions and can never be drawn completely.
Notation: segment \( \overline{ST} \) (or \( \overline{TS} \)); line \( \overleftrightarrow{ST} \) (or a small letter like m).
In simple words: A segment stops at two points and can be measured; a line never stops.

Teacher's Note:
1. A bar for a segment, a double arrow for a line.
2. Learn the symbols with the definitions.

Question 3. A ray starts at the point D and passes through the point P.
(i) Write the ray in symbols and explain what the word “ray” means here.
(ii) Explain why \( \overrightarrow{DP} \) and \( \overrightarrow{PD} \) are not the same ray.
Answer:
(i) The ray is \( \overrightarrow{DP} \). It is a part of a line that starts at D and goes on endlessly in one direction, passing through P, like the beam from a torch.
(ii) \( \overrightarrow{DP} \) starts at D and goes towards P, while \( \overrightarrow{PD} \) starts at P and goes the opposite way. Their starting points are different, so they are different rays.
In simple words: A ray starts at its first letter and goes on forever one way, so swapping the letters changes the ray.

Teacher's Note:
1. The starting point is written first.
2. Draw both rays to see they point opposite ways.

Question 4. Two rays \( \overrightarrow{OP} \) and \( \overrightarrow{OM} \) start from the same point O and form an angle.
(i) Name the vertex and the two arms of this angle.
(ii) Write two correct names for this angle in symbols and state the rule you used about the middle letter.
Answer:
(i) Vertex: O. Arms: \( \overrightarrow{OP} \) and \( \overrightarrow{OM} \).
(ii) ∠POM or ∠MOP. The vertex letter O is always written in the middle, with one point from each arm on either side.
In simple words: The corner is O, the rays are the arms, and O goes in the middle of the name.

Teacher's Note:
1. The middle letter is always the vertex.
2. Either arm point may come first.

Question 5. A full turn is divided into 360 equal parts.
(i) What is the measure of one such part called, and how is it written?
(ii) Using the full turn, find the measure of a half turn and of a quarter turn in degrees.
Answer:
(i) One part is called one degree, written 1°. The measure of an angle is the number of 1° parts inside it.
(ii) Half turn = \( \frac{1}{2} \) × 360° = 180°. Quarter turn = \( \frac{1}{4} \) × 360° = 90°.
In simple words: A full turn is 360°, so half of it is 180° and a quarter is 90°.

Teacher's Note:
1. The small circle ° stands for degrees.
2. Halve 360 to get 180, halve again to get 90.

Question 6. Write the degree range of an acute angle, an obtuse angle and a reflex angle. For each type, also give one example of a measure that belongs to it.
Answer:
Acute angle: more than 0° and less than 90°; example 35°.
Obtuse angle: more than 90° and less than 180°; example 125°.
Reflex angle: more than 180° and less than 360°; example 240°.
In simple words: Acute is below 90°, obtuse is between 90° and 180°, reflex is between 180° and 360°.

Teacher's Note:
1. The boundaries 90°, 180°, 360° are not included.
2. Any value inside the range is a correct example.

Question 7. Show that a straight angle contains exactly two right angles. Also explain what perpendicular lines are, and give one example of perpendicular lines from your classroom.
Answer:
Take a straight angle ∠AOB and fold the paper at O so that \( \overrightarrow{OB} \) falls exactly on \( \overrightarrow{OA} \).
The crease \( \overrightarrow{OC} \) splits the straight angle into two angles that match exactly, so they are equal. Each is a right angle, and 90° + 90° = 180°.
Two lines that meet at right angles are called perpendicular lines.
Example: the two edges at a corner of the blackboard.
In simple words: Folding a straight angle in half gives two equal 90° angles, and lines meeting at 90° are perpendicular.

Teacher's Note:
1. Folding shows the two halves are equal.
2. Door edges and window frames are other examples.

Question 8. Describe, in steps, how you would compare two angles drawn on two separate sheets of tracing paper by superimposition. What do you conclude if the vertices and both the arms lie exactly on top of each other?
Answer:
Step 1: Place one sheet over the other so that the vertex of the first angle falls exactly on the vertex of the second.
Step 2: Turn the top sheet until one arm of the first angle lies along one arm of the second.
Step 3: Look at the remaining arms. The angle whose second arm lies outside the other is the greater angle.
If the vertices and both pairs of arms lie exactly on top of each other, the two angles are equal.
In simple words: Put vertex on vertex and arm on arm; if the other arms match too, the angles are equal.

Teacher's Note:
1. Tracing paper lets you see both angles.
2. Arm lengths do not matter in this test.

Question 9. Rani makes an angle with two short paper straws joined by a paper clip. She then replaces the straws by two much longer straws without rotating the clip at all.
(i) Does the angle become larger? Give a reason.
(ii) If a slit is cut in a cardboard exactly to the shape of the first pair of arms, will the new longer pair of arms pass through the same slit? Explain.
Answer:
(i) No, the angle stays the same. The size of an angle is the amount of turning between the arms about the vertex. The clip was not turned, so there is no extra turning; longer arms only make the picture bigger.
(ii) Yes. Passing through the slit depends only on the angle between the arms, not on their length, and the angle has not changed.
In simple words: Longer straws do not change the angle, so they still fit the same slit.

Teacher's Note:
1. Angle = amount of turn, not arm length.
2. Same angle, same slit.

Question 10. The centre of a clock face is the vertex of the angle made by its two hands.
(i) Find the angle between the hands at 4 o’clock and write its type.
(ii) Find the smaller angle between the hands at 10 o’clock and write its type.
Answer:
The 12 hour marks divide 360° into 12 equal gaps, so one gap = 360° ÷ 12 = 30°.
(i) At 4 o’clock the hands are 4 gaps apart: 4 × 30° = 120°, an obtuse angle.
(ii) At 10 o’clock the hands are 2 gaps apart (from 10 to 12): 2 × 30° = 60°, an acute angle.
In simple words: Each hour gap is 30°, so 4 gaps give 120° and 2 gaps give 60°.

Teacher's Note:
1. Count the gaps the short way round.
2. At 10 o’clock it is 2 gaps, not 10.

Question 11. ∠AOB is a straight angle and the ray \( \overrightarrow{OC} \) divides it into ∠AOC and ∠COB. If ∠AOC = 65°, find ∠COB and write the type of each of these two angles. Give a reason for your working.
Answer:
A straight angle measures 180°, and \( \overrightarrow{OC} \) splits it into two parts, so the two parts add up to 180°.
∠COB = 180° − 65° = 115°.
∠AOC = 65° is less than 90°, so it is acute.
∠COB = 115° is between 90° and 180°, so it is obtuse.
In simple words: The two parts of a straight angle add to 180°, so the other part is 115°.

Teacher's Note:
1. Write the reason before the subtraction.
2. One part acute means the other is obtuse here.

Question 12. While using a protractor, Imran finds that one arm of an angle passes through the mark 25 and the other arm passes through the mark 110 on the same scale, but the vertex is correctly placed at the centre.
(i) Find the measure of the angle.
(ii) Explain why subtraction had to be used here.
Answer:
(i) The angle holds all the 1° parts between the marks 25 and 110, so it measures 110° − 25° = 85°.
(ii) The first arm did not pass through 0. Counting starts from 0, so when the starting reading is 25 we must take away those 25 parts from the larger reading.
In simple words: When neither arm is on 0, subtract the two readings: 110 − 25 = 85.

Teacher's Note:
1. Both readings must be on the same scale.
2. This is like measuring with a broken ruler.

Question 13. Raju measures an angle that is clearly smaller than a right angle, but he writes its measure as 140°. Explain the mistake he has made with the protractor scale, write the correct measure, and name the type of the angle.
Answer:
Raju read the wrong scale. A protractor has an inner and an outer scale, and he read the one that does not start from 0 on his base arm.
Correct measure = 180° − 140° = 40°.
40° is less than 90°, so it is an acute angle, which agrees with how the angle looks.
In simple words: He read the other scale; the real angle is 180° − 140° = 40°, which is acute.

Teacher's Note:
1. Always check if the reading fits the look of the angle.
2. The two scale readings add up to 180°.

Question 14. A round roti is cut along 8 straight cuts from the centre so that all the pieces are equal.
(i) Find the angle at the centre of one piece.
(ii) Find the angle at the centre made by three such pieces placed side by side, and write its type.
Answer:
(i) The 8 cuts divide 360° into 8 equal angles: 360° ÷ 8 = 45°.
(ii) Three pieces: 3 × 45° = 135°. It is between 90° and 180°, so it is an obtuse angle.
In simple words: Each piece is 45°, and three pieces make 135°, which is obtuse.

Teacher's Note:
1. Equal pieces share 360°.
2. Multiply for several pieces.

Question 15. Two rays \( \overrightarrow{QP} \) and \( \overrightarrow{QR} \) make an angle such that ∠PQR = 110° when it is measured on one side.
(i) Find the measure of the reflex angle made by the same two rays on the other side.
(ii) Write the type of each of the two angles and give a reason.
Answer:
(i) The two angles together make a full turn, so the reflex angle = 360° − 110° = 250°.
(ii) 110° is between 90° and 180°, so it is obtuse. 250° is between 180° and 360°, so it is reflex.
In simple words: The two sides of a pair of rays add up to 360°, so the other side is 250°.

Teacher's Note:
1. Reflex = 360° − smaller angle.
2. Mark the reflex side with a curve going the long way.

Question 16. An angle of 120° is bisected by a crease, and then one of the two parts is bisected again.
(i) Write the measure of the angle after the first bisection and after the second bisection.
(ii) Explain in your own words what an angle bisector is.
Answer:
(i) First bisection: 120° ÷ 2 = 60°. Second bisection: 60° ÷ 2 = 30°.
(ii) An angle bisector is the line or ray through the vertex that cuts an angle into two equal angles. When folded along it, the two parts fall exactly on each other.
In simple words: Halving 120° gives 60°, and halving again gives 30°.

Teacher's Note:
1. Bisect means halve.
2. Folding is a quick check for equal parts.

Question 17. A certain acute angle measures a whole number of degrees that is a multiple of 10. When its measure is doubled the answer is still an acute angle, but when its measure is tripled the answer is an obtuse angle. Find the measure of the angle and show why no other multiple of 10 fits.
Answer:
Acute multiples of 10: 10°, 20°, 30°, 40°, 50°, 60°, 70°, 80°.
Doubled must be acute (below 90°): 2 × 50° = 100° is already obtuse, so 50° and larger fail.
Tripled must be obtuse (above 90°): 3 × 30° = 90° is a right angle, and 3 × 20° = 60°, 3 × 10° = 30° are acute, so these fail.
Only 40° is left: 2 × 40° = 80° (acute) and 3 × 40° = 120° (obtuse).
In simple words: Only 40° works: doubled it is 80° (acute) and tripled it is 120° (obtuse).

Teacher's Note:
1. Test every multiple of 10 in a list.
2. A right angle is neither acute nor obtuse.

Question 18. A decorated wheel has 18 spokes fixed at equal gaps around its centre.
(i) Find the angle between two spokes that are next to each other.
(ii) Find the largest acute angle that can be formed between two spokes of this wheel, and show your reasoning.
Answer:
(i) 18 equal gaps share 360°: 360° ÷ 18 = 20°.
(ii) Angles between spokes are whole numbers of gaps: 20°, 40°, 60°, 80°, 100°, … An acute angle must be less than 90°, and 100° is already obtuse.
So the largest acute angle is 4 × 20° = 80°.
In simple words: Each gap is 20°, and the biggest multiple of 20° below 90° is 80°.

Teacher's Note:
1. List the multiples of the gap angle.
2. Stop before reaching 90°.

Question 19. At half past three, the minute hand of a clock points at 6 and the hour hand has moved half way from 3 towards 4. Find the angle between the two hands at this time and write its type. Show each step of your working.
Answer:
One hour gap on the clock is 30°.
The minute hand at 6 has turned 6 × 30° = 180° from 12.
The hour hand has turned 3 gaps and half a gap: 3 × 30° + 15° = 105° from 12.
Angle between the hands = 180° − 105° = 75°.
75° is less than 90°, so it is an acute angle.
In simple words: The minute hand is at 180° and the hour hand at 105°, so they are 75° apart.

Teacher's Note:
1. The hour hand moves too, not just the minute hand.
2. Half a gap is 15°.

Question 20. A toy car is facing east. It first makes a quarter turn clockwise, then a half turn, and finally one more quarter turn clockwise.
(i) In which direction is the car facing at the end?
(ii) Find the total amount of turning in degrees and name the angle that this total turn makes.
Answer:
(i) East → a quarter turn clockwise → south. A half turn → north. One more quarter turn clockwise → east. The car faces east again.
(ii) Total turning = 90° + 180° + 90° = 360°. This is one full turn, a complete angle, which is why the car faces its starting direction.
In simple words: The turns add up to 360°, a full turn, so the car ends up facing east again.

Teacher's Note:
1. Draw the four directions and follow the turns.
2. A full turn brings you back where you started.