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02
NCERT Solutions · Maths Ch 2
Lines and Angles
48 questions · 6 exercises
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Ex 2.1
Ex 2.2
Ex 2.3
Ex 2.4
Ex 2.5
Ex 2.6
NCERT answers with full exam-style working, step by step. Tap a question to open its solution.
1
EXERCISE 2.1 · 6 QUESTIONS
Points, Lines, Rays
Q1
Rihan marked a point on a piece of paper. How many lines can he draw that pass through the point? Sheetal marked two points. How many different lines can she draw that pass through both of the points?
(2 marks)
Q2
Name the line segments in Fig. 2.4. Which of the five marked points are on exactly one of the line segments? Which are on two of the line segments? (The five marked points are L, M, P, Q, R joined one after another.)
(2 marks)
Q3
Name the rays shown in Fig. 2.5. Is T the starting point of each of these rays? (The figure shows points A, T, N and B.)
(2 marks)
Q4
Draw a rough figure and write labels appropriately to illustrate each of the following: (a) OP and OQ meet at O. (b) XY and PQ intersect at point M. (c) Line l contains points E and F but not point D. (d) Point P lies on AB.
(3 marks)
Q5
In Fig. 2.6, name: (a) Five points (b) A line (c) Four rays (d) Five line segments (The figure shows a line through D, E, O, B with ray OC.)
(2 marks)
Q6
Here is a ray OA (Fig. 2.7). It starts at O and passes through the point A. It also passes through the point B. (a) Can you also name it as OB? Why? (b) Can we write OA as AO? Why or why not?
(2 marks)
2
EXERCISE 2.2 · 6 QUESTIONS
Angles
Q7
Can you find the angles in the given pictures? Draw the rays forming any one of the angles and name the vertex of the angle. (The figure has points A, B, C, D.)
(2 marks)
Q8
Draw and label an angle with arms ST and SR.
(2 marks)
Q9
Explain why ∠APB cannot be labelled as ∠P. (The figure shows P as a vertex with points A, B and C around it.)
(2 marks)
Q10
Name the angles marked in the given figure. (The figure has points P, Q, R, T.)
(1 mark)
Q11
Mark any three points on your paper that are not on one line. Label them A, B, C. Draw all possible lines going through pairs of these points. How many lines do you get? Name them. How many angles can you name using A, B, C? Write them down.
(2 marks)
Q12
Now mark any four points on your paper so that no three of them are on one line. Label them A, B, C, D. Draw all possible lines going through pairs of these points. How many lines do you get? Name them. How many angles can you name using A, B, C, D? Write them all down.
(3 marks)
3
EXERCISE 2.3 · 7 QUESTIONS
Comparing Angles and Right Angles
Q13
Fold a rectangular sheet of paper, then draw a line along the fold created. Name and compare the angles formed between the fold and the sides of the paper. Make different angles by folding the sheet and compare the angles. Which is the largest and smallest angle you made?
(2 marks)
Q14
In each case, determine which angle is greater and why: (a) ∠AOB or ∠XOY (b) ∠AOB or ∠XOB (c) ∠XOB or ∠XOC
(2 marks)
Q15
Which angle is greater: ∠XOY or ∠AOB? Give reasons.
(2 marks)
Q16
Get a slanting crease on the paper. Now, try to get another crease that is perpendicular to the slanting crease. (a) How many right angles do you have now? Justify why the angles are exact right angles. (b) Describe how you folded the paper so that any other person who does not know the process can follow it to get the right angle.
(3 marks)
Q17
How many right angles do the windows of your classroom contain? Do you see other right angles in your classroom?
(2 marks)
Q18
Join A to other grid points in the figure by a straight line to get a straight angle. What are all the different ways of doing it?
(1 mark)
Q19
Now join A to other grid points in the figure by a straight line to get a right angle. What are all the different ways of doing it? [Hint: Extend the line further. To get a right angle at A, we need to draw a line through it that divides the straight angle CAB into two equal parts.]
(2 marks)
4
EXERCISE 2.4 · 3 QUESTIONS
Acute and Obtuse
Q20
Identify acute, right, obtuse and straight angles in the previous figures. Make a few acute angles and a few obtuse angles. Draw them in different orientations.
(2 marks)
Q21
Do you know what the words acute and obtuse mean? Acute means sharp and obtuse means blunt. Why do you think these words have been chosen?
(1 mark)
Q22
Find out the number of acute angles in each of the figures below. What will be the next figure and how many acute angles will it have? Do you notice any pattern in the numbers? (The figures are a sequence of triangles with 0, 1, 2 inner triangles...)
(2 marks)
5
EXERCISE 2.5 · 13 QUESTIONS
Measuring Angles
Q23
Write the measures of the angles (from a unit-circle figure where 1 unit = 1°): (a) ∠KAL (b) ∠WAL (c) ∠TAK
(2 marks)
Q24
Name the different angles in the figure and write their measures. (Rays OP, OQ, OR, OS, OT, OU from O.)
(2 marks)
Q25
In Fig. 2.20, we have ∠AOB = ∠BOC = ∠COD = ∠DOE = ∠EOF = ∠FOG = ∠GOH = ∠HOI = ____ . Why?
(2 marks)
Q26
Find the degree measures of the following angles using your protractor. (Three angles with vertex H: ∠IHJ in each of the three figures.)
(2 marks)
Q27
Find the degree measures of the angles given (two angles with vertex H and arms HI and HJ). Check if your paper protractor can be used here!
(2 marks)
Q28
How can you find the degree measure of the angle given below using a protractor? (The marked angle is a reflex angle drawn with a curve going the long way round.)
(2 marks)
Q29
Measure and write the degree measures for each of the following angles: (a) to (f) (six angles drawn in different positions).
(2 marks)
Q30
Find the degree measures of ∠BXE, ∠CXE, ∠AXB and ∠BXC.
(2 marks)
Q31
Find the degree measures of ∠PQR, ∠PQS and ∠PQT.
(2 marks)
Q32
Angles in a clock: (a) The hands of a clock make different angles at different times. At 1 o'clock, the angle between the hands is 30°. Why? (b) What will be the angle at 2 o'clock? And at 4 o'clock? 6 o'clock? (c) Explore other angles made by the hands of a clock.
(3 marks)
Q33
The angle of a door: Is it possible to express the amount by which a door is opened using an angle? What will be the vertex of the angle and what will be the arms of the angle?
(2 marks)
Q34
Vidya is enjoying her time on the swing. She notices that the greater the angle with which she starts the swinging, the greater is the speed she achieves on her swing. But where is the angle? Are you able to see any angle?
(2 marks)
Q35
A toy has slanting slabs attached to its sides; the greater the angles or slopes of the slabs, the faster the balls roll. Can angles be used to describe the slopes of the slabs? What are the arms of each angle? Which arm is visible and which is not?
(2 marks)
6
EXERCISE 2.6 · 13 QUESTIONS
Drawing and Types of Angles
Q36
In Fig. 2.23, list all the angles possible. Did you find them all? Now guess the measures of all the angles. Then measure the angles with a protractor and record all your numbers in a table.
(2 marks)
Q37
Use a protractor to draw angles having the following degree measures: (a) 110° (b) 40° (c) 75° (d) 112° (e) 134°
(2 marks)
Q38
Draw an angle whose degree measure is the same as the angle given below (angle IHJ). Also, write down the steps you followed to draw the angle.
(2 marks)
Q39
In each of the grids, join A to other grid points in the figure by a straight line to get: (a) an acute angle (b) an obtuse angle (c) a reflex angle. Mark the intended angles with curves.
(2 marks)
Q40
Use a protractor to find the measure of each angle. Then classify each angle as acute, obtuse, right, or reflex. (a) ∠PTR (b) ∠PTQ (c) ∠PTW (d) ∠WTP
(3 marks)
Q41
In this figure, ∠TER = 80°. What is the measure of ∠BET? What is the measure of ∠SET? (B, E, R are on a straight line; ES is perpendicular to BR, with T between S and R.)
(2 marks)
Q42
Draw angles with the following degree measures: (a) 140° (b) 82° (c) 195° (d) 70° (e) 35°
(2 marks)
Q43
Estimate the size of each angle and then measure it with a protractor (angles a to f). Classify these angles as acute, right, obtuse or reflex angles.
(2 marks)
Q44
Make any figure with three acute angles, one right angle and two obtuse angles.
(2 marks)
Q45
Draw the letter 'M' such that the angles on the sides are 40° each and the angle in the middle is 60°.
(2 marks)
Q46
Draw the letter 'Y' such that the three angles formed are 150°, 60° and 150°.
(2 marks)
Q47
The Ashoka Chakra has 24 spokes. What is the degree measure of the angle between two spokes next to each other? What is the largest acute angle formed between two spokes?
(2 marks)
Q48
Puzzle: I am an acute angle. If you double my measure, you get an acute angle. If you triple my measure, you will get an acute angle again. If you quadruple (four times) my measure, you will get an acute angle yet again! But if you multiply my measure by 5, you will get an obtuse angle measure. What are the possibilities for my measure?
(3 marks)