CBSE Class 6 Mathematics Lines and Angles Worksheet Set 01

Long Answer Questions

Teacher's Note:
1. Arrow heads show the endless directions.
2. Link the number of arrows with the number of endless directions.

Question 2. An angle is formed by two rays \( \overrightarrow{BD} \) and \( \overrightarrow{BE} \) that start from the same point B.
(i) Draw a rough figure of this angle, mark the small curve at the corner and label the vertex and the two arms.
(ii) Write all the correct names of this angle in symbols and state the rule about the position of the vertex letter in the name.
(iii) Three rays \( \overrightarrow{PA} \), \( \overrightarrow{PB} \) and \( \overrightarrow{PC} \) start from the same point P. Explain why one of the angles in this figure should not be called simply ∠P.
Answer:
(i) Mark the point B and draw two rays from it, one through D and one through E. Draw a small curve between the arms near B. B is the vertex; \( \overrightarrow{BD} \) and \( \overrightarrow{BE} \) are the arms.

(ii) ∠DBE or ∠EBD. The vertex letter is always in the middle, with one point from each arm on either side. When there is no chance of confusion, it may also be called ∠B.
(iii) With three rays from P there are three different angles at P: ∠APB, ∠BPC and ∠APC.
The name ∠P does not tell which pair of arms is meant, so it is confusing. Using one point from each arm with the vertex names the angle exactly.
In simple words: Put the vertex letter in the middle, and use three letters whenever more than two rays meet at a point.

Teacher's Note:
1. ∠B is allowed only when it is clear.
2. Count the angles when three rays meet: there are three.

Question 3. Vidya opens the cover of her book in different ways, and each opening makes an angle at the spine of the book.
(i) Describe what is meant by the size of an angle, using the idea of turning.
(ii) Explain, using the book cover, what a quarter turn, a half turn and a full turn mean, and write the degree measure of each.
(iii) Name the angle formed in each of these three cases and state how many right angles make one straight angle and how many make one full turn.
Answer:
(i) The size of an angle is the amount of turn about the vertex needed to bring the first arm onto the second arm. The more the cover is turned, the greater the angle; the length of the arms makes no difference.
(ii) Quarter turn = \( \frac{1}{4} \) × 360° = 90°; the cover stands upright like the letter L.
Half turn = \( \frac{1}{2} \) × 360° = 180°; the cover lies flat on the table and the two arms are in one straight line.
Full turn = 360°; the cover goes right round back to its starting position.
(iii) A quarter turn gives a right angle, a half turn a straight angle and a full turn a complete (whole) angle.
Two right angles make a straight angle (90° + 90° = 180°), and four right angles make a full turn (4 × 90° = 360°).
In simple words: Quarter, half and full turns are 90°, 180° and 360°: right, straight and complete angles.

Teacher's Note:
1. Act the turns out with a real book.
2. Remember 90, 180, 360 together.

Question 4. Here is a list of angle measures:
28°, 90°, 118°, 180°, 265°, 360°
(i) Classify each measure as acute, right, obtuse, straight, reflex or a complete (full) angle, and give a short reason for each.
(ii) Write the degree range of an acute angle, an obtuse angle and a reflex angle.
(iii) Rohit says, “Every angle that is bigger than a right angle must be an obtuse angle.” Is he correct? Justify your answer with an example.
Answer:
(i) 28°: acute, as it is more than 0° and less than 90°.
90°: right angle, exactly a quarter turn.
118°: obtuse, as it is between 90° and 180°.
180°: straight angle, exactly a half turn.
265°: reflex, as it is between 180° and 360°.
360°: complete angle, one full turn.
(ii) Acute: more than 0° and less than 90°. Obtuse: more than 90° and less than 180°. Reflex: more than 180° and less than 360°.
(iii) No. An angle bigger than a right angle is obtuse only while it is below 180°. For example, 265° is bigger than 90° but is a reflex angle, and 180° is a straight angle, not obtuse.
In simple words: Compare each angle with 90°, 180° and 360°; not every angle above 90° is obtuse.

Teacher's Note:
1. Exact values 90°, 180°, 360° have their own names.
2. One example is enough to show Rohit is wrong.

Question 5. Your friend has never used a protractor before.
(i) Write the steps you would follow to measure ∠ABC with a protractor, from placing the protractor to reading the final number.
(ii) Explain why a protractor carries two sets of numbers and how you decide which set to read.
(iii) Write any two mistakes a student may make while using a protractor, and state how each mistake can be corrected.
Answer:
(i) Step 1: Place the protractor so that its centre mark is exactly on the vertex B.
Step 2: Turn it so that the arm \( \overrightarrow{BC} \) lies along the base line through the 0 mark.
Step 3: Use the scale that starts at 0 on this arm and read the number through which \( \overrightarrow{BA} \) passes.
Step 4: That number is the measure of ∠ABC in degrees.
(ii) One scale rises from left to right and the other from right to left, so an angle can be measured whichever side the base arm lies on. Always read the scale whose 0 is on the base arm.
(iii) Mistake 1: the vertex is not on the centre mark, so the reading is wrong. Correction: slide the protractor until the centre sits exactly on the vertex.
Mistake 2: reading the other scale, for example 140° instead of 40°. Correction: count from the 0 that lies on the base arm.
(Another mistake: the arms are too short to reach the scale. Correction: extend them with a ruler; this does not change the angle.)
In simple words: Put the centre on the vertex, one arm on 0, and read the same scale where the other arm crosses.

Teacher's Note:
1. Check the reading against the look of the angle.
2. Extending arms is always allowed.

Question 6. Answer the following about drawing angles with a protractor.
(i) Write the steps to draw ∠MAT = 125°, taking \( \overrightarrow{AT} \) as the base arm, and name the type of this angle.
(ii) A reflex angle cannot be read directly on an ordinary half-circle protractor. Explain how you would draw a reflex angle of 250° using such a protractor.
(iii) Why must the centre point of the protractor be placed exactly on the vertex while drawing?
Answer:
(i) Step 1: Draw the base ray \( \overrightarrow{AT} \) with a ruler.
Step 2: Place the centre of the protractor on A and line up \( \overrightarrow{AT} \) with the 0 line.
Step 3: From that 0, count 10, 20, 30, … up to 125 on the same scale and mark a point M.
Step 4: Join A to M. Then ∠MAT = 125°. It lies between 90° and 180°, so it is an obtuse angle.
(ii) The two angles on the two sides of the arms make a full turn, so 360° − 250° = 110°. Draw an angle of 110° in the usual way and mark the curve on the other side of the arms. That marked angle is the reflex angle of 250°.
(iii) All the degree marks are measured from the centre of the protractor. If the centre is not on the vertex, the arm does not cross the correct mark and the angle drawn is wrong.
In simple words: Draw the base arm, put the centre on the vertex, count to the angle, and for a reflex angle draw 360° minus it.

Teacher's Note:
1. Count on the scale whose 0 is on the base arm.
2. Mark the reflex side clearly with a curve.

Question 7. Sita makes her own protractor by folding a paper circle.
(i) Describe the folds she must make, step by step, to get the marks 180°, 90° and 45° on her semicircle, and write the fraction of the full turn each one is.
(ii) One more half fold is made after the 45° crease. Find the measure of the smallest angle now formed between two neighbouring creases, and show the working.
(iii) What is an angle bisector? Name the angle bisector of the straight angle in Sita’s semicircle.
Answer:
(i) A paper circle stands for one full turn, 360°.
Fold it into two equal halves and cut along the crease. The semicircle is \( \frac{1}{2} \) of the full turn, that is 180°; write 0° at one bottom corner and 180° at the other.
Fold the semicircle in half. This crease is \( \frac{1}{4} \) of the full turn, that is \( \frac{360^\circ}{4} \) = 90°; mark 90° at the top.
Fold once more to halve the right angle. This crease is \( \frac{1}{8} \) of the full turn, that is \( \frac{360^\circ}{8} \) = 45°; the matching crease on the other side is 180° − 45° = 135°.
(ii) After one more half fold, the straight angle is divided into 8 equal parts, so the smallest angle is \( \frac{180^\circ}{8} \) = 22.5° (half of 45°).
(iii) An angle bisector is the line or ray through the vertex that cuts an angle into two equal angles. In the semicircle, the crease marked 90° is the bisector of the straight angle.
In simple words: Each fold halves the angle: 180°, 90°, 45°, then 22.5°.

Teacher's Note:
1. Each fold halves the previous angle.
2. 22.5° is half of 45°.

Question 8. The figure shows five rays \( \overrightarrow{OA} \), \( \overrightarrow{OB} \), \( \overrightarrow{OC} \), \( \overrightarrow{OD} \) and \( \overrightarrow{OE} \) drawn from the same point O, with A, O and E in one straight line.
(i) Measure ∠AOB, ∠AOC and ∠AOD with your protractor and write each measure.
(ii) Classify each of these three angles as acute, right or obtuse, and give a reason for each.
(iii) Find ∠DOE without measuring it, and explain the idea you used.
Answer:
(i) ∠AOB = 40°, ∠AOC = 90° and ∠AOD = 145°.
(ii) ∠AOB = 40° is acute, as it is less than 90°.
∠AOC = 90° is a right angle, exactly a quarter turn; \( \overrightarrow{OC} \) is perpendicular to the line through A and E.
∠AOD = 145° is obtuse, as it is between 90° and 180°.
(iii) A, O, E are in one straight line, so ∠AOE is a straight angle of 180°, and \( \overrightarrow{OD} \) splits it into ∠AOD and ∠DOE.
∠DOE = 180° − 145° = 35°, an acute angle. (In the same way, ∠BOE = 180° − 40° = 140°.)
In simple words: Measure from OA, then use 180° along the straight line to find ∠DOE = 35°.

Teacher's Note:
1. Read every angle from the 0 on OA.
2. Parts of a straight angle add up to 180°.

Question 9. Angles can be compared in many ways.
(i) A pair of rotating arms is made with two straws and a paper clip. Explain why replacing the straws by longer straws does not change the angle between them.
(ii) Describe the “slit in the cardboard” test, and state what it tells us about two pairs of rotating arms that pass through the same slit.
(iii) Describe how a transparent circular sheet can be used to compare two angles that are drawn on two different pages, without cutting or tracing either page.
Answer:
(i) An angle is the amount of turning between its arms about the vertex. Changing the straws without turning the clip adds no turning, so the angle is unchanged; only the picture looks bigger. That is why we may extend arms before measuring.
(ii) Cut a slit in a cardboard by tracing one pair of rotating arms. Slide other pairs of arms into the slit without turning them. If an angle is bigger or smaller than the slit angle, the arms do not go through; only a pair with exactly the same angle passes. So two pairs that pass through the same slit have equal angles, whatever the length of their arms.
(iii) Put the centre of the transparent circle on the vertex of the first angle and mark the points A and B where the arms cross the edge of the circle.
Move the sheet to the second angle, put the centre on its vertex and turn it until one arm passes through A. If the other arm falls inside the marked part, the second angle is smaller; if outside, it is greater; if it passes through B, the two angles are equal.
In simple words: Angles depend only on turning, so we can compare them by slits or a see-through circle, not by arm length.

Teacher's Note:
1. All three methods ignore arm length.
2. Each method puts vertex on vertex.

Question 10. Angles are hidden in many everyday actions.
(i) A door is pushed open from the wall. Name the vertex and the two arms of the angle formed, and state the measure of the angle when the door lies flat along the wall.
(ii) A child on a swing starts from the lowest position and rises to one side. Explain where the vertex and the arms of the angle are in this case.
(iii) A toy has slanting slabs on its sides, and balls roll faster down a steeper slab. Explain how an angle describes the slope, and state which arm of this angle is usually not drawn.
Answer:
(i) The vertex is where the door is hinged to the wall. The arms are the edge of the open door and the edge of the wall from which it turned. When the door is pushed right back flat along the wall, its arms lie in one straight line, so the angle is a straight angle of 180°.
(ii) The vertex is the point at the top where the ropes are tied. One arm is the rope in its lowest, hanging position, and the other is the rope when the child has risen to one side. The angle is the turn between these two positions.
(iii) One arm is the slanting edge of the slab and the other is the horizontal line at its foot; they meet at the vertex. A steeper slab makes a bigger angle with the horizontal, so balls roll faster. The horizontal arm is usually not drawn and has to be imagined.
In simple words: The hinge, the top of the swing and the foot of the slab are vertices; the moving parts and the hidden horizontal are arms.

Teacher's Note:
1. Find the point that does not move: it is the vertex.
2. Some arms have to be imagined.

Case Study Based Questions

Question 11. Meera opens her new geometry box and takes out the protractor. It is a half circle made of clear plastic. Her teacher tells the class that the straight angle at the centre of this half circle has been divided into 180 equal parts, and each part is one degree. There is a small mark at the middle of the straight edge; this is the centre of the protractor. Along the curved edge there are two sets of numbers, one rising from right to left and the other rising from left to right. Meera places the centre on the corner of an angle, keeps one arm along the 0 line on the right, and sees that the second arm passes through 65 on the scale that starts from 0 on the right.
(i) Into how many equal parts is the straight angle of a protractor divided?
(ii) On which point of the angle must the centre mark of the protractor be placed?
(iii) Why does a protractor carry two sets of numbers?
(iv) Write the measure of Meera’s angle and the type of angle it is.
Answer:
(i) Into 180 equal parts, each of 1°.
(ii) On the vertex, the corner where the two arms meet.
(iii) So that an angle can be measured from either side: one scale starts from 0 on the right and the other from 0 on the left. We read the scale whose 0 lies on the base arm.
(iv) Meera’s angle is 65°. It is less than 90°, so it is an acute angle.
In simple words: A protractor splits 180° into 1° parts; Meera’s 65° angle is acute.

Teacher's Note:
1. Centre on vertex, base arm on 0.
2. Use the scale whose 0 is on the base arm.

Question 12. A round wall clock hangs above the blackboard of Class VI-B. The twelve numbers on its face are written at equal gaps all around the centre. The two hands of the clock meet at the centre, so the centre is the vertex of the angle and the hands are its arms. As time passes, the hands turn and the angle between them keeps changing. The teacher asks the class to look at the clock at different hours of the day and to say what kind of angle the hands make. The children notice that at some hours the two hands look like the letter L and at some other hours they lie in one straight line.
(i) Find the angle turned between two numbers that are next to each other on the clock face.
(ii) Find the angle between the hands at 2 o’clock and name its type.
(iii) At which hour do the two hands form a straight angle? Write its measure.
(iv) At 9 o’clock, find the reflex angle between the two hands.
Answer:
(i) The 12 numbers divide 360° into 12 equal parts: 360° ÷ 12 = 30°.
(ii) At 2 o’clock the hands are 2 gaps apart: 2 × 30° = 60°, an acute angle.
(iii) At 6 o’clock: 6 × 30° = 180°, a straight angle.
(iv) At 9 o’clock the hands are 3 gaps apart (9 to 12), so the smaller angle is 3 × 30° = 90°. The reflex angle is 360° − 90° = 270°.
In simple words: Each hour gap is 30°; count the gaps and use 360° for the reflex side.

Teacher's Note:
1. The L shape at 3 and 9 o’clock is a right angle.
2. Reflex = 360° − smaller angle.

Question 13. A sketch of Raju’s village is drawn on the blackboard. The post office is marked as the point P, the school as the point Q and the old well as the point R. A straight road joins P to Q and is the shortest path between them. From the top of the water tank at R, a strong searchlight is switched on at night; its beam travels straight and seems to go on and on without ending. The teacher reminds the children that a point only shows a position and has no length, breadth or height, and that every point on the sketch is named by a capital letter.
(i) Which geometrical idea does the straight road from P to Q stand for? Write it in symbols.
(ii) How many different lines can be drawn that pass through both P and Q?
(iii) Which geometrical idea does the searchlight beam stand for, and which point is its starting point?
(iv) How many lines can be drawn through the single point R?
Answer:
(i) A line segment, the shortest path joining two points: \( \overline{PQ} \) (or \( \overline{QP} \)).
(ii) Only one line, because two points fix exactly one line.
(iii) A ray, as it starts at one point and goes on endlessly in one direction. Its starting point is R; it can be written \( \overrightarrow{RX} \) for any point X on the beam.
(iv) Countless lines, in all directions.
In simple words: The road is a segment, the beam is a ray from R, and one point has endless lines through it.

Teacher's Note:
1. Two points: one line. One point: countless lines.
2. A ray's name starts with its starting point.

Question 14. Asha is cutting coloured paper with a pair of scissors for a chart. The two blades of the scissors are joined by a small screw. When she opens the blades, the two blades form an angle at the screw. Her younger brother has a small pair of scissors with much shorter blades. Asha opens her big scissors a little and her brother opens his small scissors exactly as much, and they trace both the openings on paper and place one tracing over the other. The corner points fall on one another and both the pairs of traced arms lie exactly on top of one another.
(i) Which part of the scissors is the vertex of the angle, and which parts are its arms?
(ii) What happens to the angle when Asha opens the blades wider?
(iii) What do we call the method of placing one traced angle over the other to compare them?
(iv) Whose scissors make the greater angle? Give a reason for your answer.
Answer:
(i) The screw is the vertex, and the two blades are the arms.
(ii) Opening wider means more turning about the screw, so the angle becomes greater.
(iii) Comparing by superimposition.
(iv) Neither; the two angles are equal. On superimposition the corners and both pairs of arms lie exactly on one another. Asha’s blades are longer, but arm length does not change the size of an angle.
In simple words: The screw is the vertex; both scissors open by the same turn, so the angles are equal.

Teacher's Note:
1. The fixed joint is always the vertex.
2. Superimposition ignores arm length.

Question 15. The Ashoka Chakra in the middle of our national flag is a wheel with 24 spokes. All the spokes start from the centre of the wheel and are fixed at equal gaps all around it, so the full turn of 360° at the centre is shared equally among the 24 gaps. The children of Class VI draw a large Chakra on a chart paper for Republic Day. They number the spokes 1, 2, 3, …, 24 going round the centre, and then measure some of the angles formed between pairs of spokes with a protractor.
(i) Find the angle between two spokes that are next to each other.
(ii) Find the angle between spoke 1 and spoke 4.
(iii) Find the largest acute angle that is formed between two spokes of this Chakra.
(iv) How many gaps apart are two spokes that lie in one straight line?
Answer:
(i) 360° ÷ 24 = 15°.
(ii) From spoke 1 to spoke 4 there are 3 gaps: 3 × 15° = 45°, an acute angle.
(iii) Angles between spokes are 15°, 30°, 45°, 60°, 75°, 90°, … The largest one below 90° is 5 × 15° = 75°.
(iv) Spokes in one straight line make 180°, and 180° ÷ 15° = 12. So they are 12 gaps apart, exactly opposite each other.
In simple words: Each gap is 15°, so count gaps and multiply by 15°.

Teacher's Note:
1. Spoke 1 to spoke 4 is 3 gaps, not 4.
2. Opposite spokes make a straight angle.

Question 16. The ceiling fan in the school library has three blades. All three blades are fixed to the same round hub at the centre and are spread at equal gaps, so that the fan stays balanced while turning. Looking straight up at the switched-off fan, Nikhil sees that the three blades divide the full turn around the hub into three equal angles. In the staff room there is another fan of the same company with four blades, again fixed at equal gaps around its hub.
(i) Find the angle between two blades of the library fan that are next to each other.
(ii) Name the type of this angle and give the reason.
(iii) Find the reflex angle formed between the same two blades.
(iv) Find the angle between two neighbouring blades of the four-blade fan and name its type.
Answer:
(i) 360° ÷ 3 = 120°.
(ii) Obtuse, because 120° is more than 90° and less than 180°.
(iii) The two sides together make a full turn: 360° − 120° = 240°.
(iv) 360° ÷ 4 = 90°, a right angle.
In simple words: Three blades share 360° as 120° each; four blades share it as 90° each.

Teacher's Note:
1. Equal gaps share 360° equally.
2. Reflex = 360° − the smaller angle.

Question 17. In the craft period, every child is given a rectangular sheet of paper. The teacher asks them to make a slanting fold anywhere across the sheet and to open it out. A straight crease appears. Next, they have to fold the sheet once more so that one part of the slanting crease falls exactly on the other part of the same crease, and then open the sheet out again. Two creases are now seen, crossing each other at a point. Rina marks this point as O and marks the four angles around O. She checks each of them by placing one over the other through folding.
(i) What kind of angle does one full slanting crease make at the point O on one side?
(ii) What is the measure of each of the four angles formed around O?
(iii) Name the type of the four angles formed and write what the two creases are called.
(iv) Add the four angles around O. What total do you get, and why?
Answer:
(i) The crease is straight, so on one side it makes a straight angle of 180° at O.
(ii) The second fold halves that straight angle: 180° ÷ 2 = 90°, and the same happens on the other side.
(iii) All four are right angles. Lines meeting at right angles are perpendicular lines, so the two creases are perpendicular.
(iv) 90° + 90° + 90° + 90° = 360°. The four angles fill one complete turn around O.
In simple words: Folding a straight crease onto itself makes four right angles that add up to 360°.

Teacher's Note:
1. Folding one part of a crease onto the other halves a straight angle.
2. Angles all round a point add to 360°.

Question 18. Four straight walking paths start from a fountain in the middle of a park. The paths lead to the gate A, the rose bed B, the canteen C and the pond D. The fountain is the point O where all four paths meet, so the four paths make four angles that together fill the whole space around O. The gardener has measured three of these angles and marked them in the plan shown below. The angle between the path to the pond and the path to the gate has not been measured.
(i) What is the sum of the four angles marked around the point O?
(ii) Find the measure of ∠DOA.
(iii) Classify ∠BOC and ∠COD as acute or obtuse, with reasons.
(iv) Find the reflex angle at O on the other side of ∠AOB.
Answer:
(i) The four angles fill one full turn around O, so their sum is 360°.
(ii) ∠AOB + ∠BOC + ∠COD + ∠DOA = 360°.
∠DOA = 360° − (100° + 85° + 95°) = 360° − 280° = 80°.
(iii) ∠BOC = 85° is acute, as it is less than 90°. ∠COD = 95° is obtuse, as it is between 90° and 180°.
(iv) Reflex angle = 360° − 100° = 260°. (Check: 85° + 95° + 80° = 260°.)
In simple words: Angles around a point add to 360°, so the missing one is 80°.

Teacher's Note:
1. Add the known angles first, then subtract from 360°.
2. 85° and 95° are close but on opposite sides of 90°.

Teacher's Note:
1. A conjecture is a guess based on a pattern.
2. Use 180° to find a missing angle.

Teacher's Note:
1. The score is always the difference, bigger minus smaller.
2. Lower score wins in this game.