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Solutions

Overview

Step-by-step NCERT solutions for Lines and Angles (Chapter 2, NCERT Class 6 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Lines and Angles textbook chapter.

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What these solutions cover

All 38 questions in Lines and Angles are solved in the PDF. Here's what's inside, exercise by exercise:

Ray

  1. Rihan marked a point on a piece of paper. How many lines can he draw that pass through the point? Sheetal marked two points on a piece of paper. How many different lines can she draw that pass through both of the points? Can you help Rihan and Sheetal find their answers?
  2. 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?
  3. Name the rays shown in Fig. 2.5. Is T the starting point of each of these rays?
  4. In Fig. 2.6, name:
    • (a) Five points,
    • (b) A line,
    • (c) Four rays,
    • (d) Five line segments.
  5. 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?

Angle

  1. 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. (Pictures show a bicycle and a tiled pattern.)
  2. Name the angles marked in the given figure. (Figure shows two rays from vertex T — one toward P and one toward Q — with a curve at T, and a horizontal ray toward R.)
  3. 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, and mark each of them with a curve as in Fig. 2.9.
  4. 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, and mark each of them with a curve as in Fig. 2.9.

Comparing Angles

  1. In each case, determine which angle is greater and why.
    • (a) ∠AOB or ∠XOY,
    • (b) ∠AOB or ∠XOB,
    • (c) ∠XOB or ∠XOC. (Figure shows rays OA, OX, OY, OB, OC from vertex O, in that order from upper-left going clockwise to right, with Y between X and B, and C to the right of B on the horizontal.)
  2. Which angle is greater: ∠XOY or ∠AOB? Give reasons. (Figure shows rays OX and OA going upward-left with different amounts of spread, and OY, OB going right from vertex O.)

Special Types of Angles

  1. How many right angles do the windows of your classroom contain? Do you see other right angles in your classroom?
  2. 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? (A and B are grid points; B is to the right of A on the same row.)
  3. 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 as shown. To get a right angle at A, draw a line through it that divides the straight angle CAB into two equal parts.)
  4. 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 doesn't know the process can simply follow your description to get the right angle.

Classifying Angles

  1. Identify acute, right, obtuse and straight angles in the previous figures. (Referring to the three groups of angles shown in the 'Classifying Angles' section.)
  2. 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?
  3. Find out the number of acute angles in each of the figures below. (Three figures: a single triangle, a triangle subdivided into 4 smaller triangles, and a triangle subdivided further with 9 smaller triangles.) What will be the next figure and how many acute angles will it have? Do you notice any pattern in the numbers?

Measuring Angles — Unlabelled Protractor

  1. Write the measures of the following angles using the unlabelled protractor shown:
    • (a) ∠KAL,
    • (b) ∠WAL,
    • (c) ∠TAK. (The vertex A coincides with the centre; L is at 0° (rightmost base), K is 30° from L, W is 50° from L, T is at 150° from L measured going left, or equivalently at 120° from K on the same side.)

Measuring Angles — Make Your Own Protractor

  1. In Step 4 of making the protractor: fold the quarter-circle in half again. This new fold is 1/8 of the circle. What is the measure of this angle? (Fill in the blank: 1/8 of 360° = _____.)
  2. In Step 5, continuing with another half fold of the 45° section, we get an angle of measure _____ (fill in the blank).
  3. Think! In Fig. 2.19, we have ∠AOB = ∠BOC = ∠COD = ∠DOE = ∠EOF = ∠FOG = ∠GOH = ∠HOI = _____. Why?
  4. Find the degree measures of the following angles using your protractor. (Three angle figures:
    • (i) ∠IHJ with H as vertex, I and J as the two arms;
    • (ii) ∠GHK (or ∠IHJ) — a narrower angle;
    • (iii) ∠IHJ — a wider angle with H as vertex.)
  5. Find the degree measures for the angles given below. Check if your paper protractor can be used here! (Two figures:
    • (i) a triangle with vertex at H, arms going to I and J;
    • (ii) an obtuse angle at H with arms toward I and J.)
  6. How can you find the degree measure of the angle given below using a protractor? (The figure shows a reflex angle — the marked (shaded) angle is greater than 180°.)
  7. Measure and write the degree measures for each of the following angles: (a), (b), (c), (d), (e), (f) — six angles shown as figures.
  8. Find the degree measures of ∠BXE, ∠CXE, ∠AXB and ∠BXC. (Figure shows a protractor placed with centre at X; rays XA, XB, XC go to the left and above-left, ray XE goes to the right along the base.)
  9. Find the degree measures of ∠PQR, ∠PQS and ∠PQT. (Figure shows vertex Q with ray QP going left and rays QR, QS, QT going upward-right at increasing angles.)

Where Are the Angles?

  1. 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?
  2. 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?
  3. 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?
  4. Here is a toy with 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?
  5. Observe the images below where there is an insect and its rotated version. Can angles be used to describe the amount of rotation? How? What will be the arms of the angle and the vertex? (Hint: Observe the horizontal line touching the insects.)

Drawing Angles

  1. 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. Record all your numbers in a table. See how close your guesses are to the actual measures. (Fig. 2.23 shows points A, P, R, B, C, D, L, S with lines joining them forming a figure.)

Types of Angles and their Measures

  1. 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. (Figure shows vertex T with rays TR, TQ going upper-right, TW going lower-right, and TP going left.)

Types of Angles and their Measures — Let's Explore and Further Problems

  1. In this figure, ∠TER = 80°. What is the measure of ∠BET? What is the measure of ∠SET? (Figure shows a straight line through B, E, R; ray ET goes upward making 80° with ER; ray ES is perpendicular to the line, making a right angle ∠BES = 90°.)
  2. 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?
  3. 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?
Sample

Worked examples

3 questions from Lines and Angles, solved step by step.

Special Types of Angles

How many right angles do the windows of your classroom contain? Do you see other right angles in your classroom?

A rectangular window has four corners. Each corner is a right angle (90°).

So each window contains 4 right angles.

Other right angles in a classroom: corners of the blackboard, corners of the door, corners of books, corners of tiles on the floor, edges where walls meet the floor or ceiling.

Answer: Each rectangular window contains 4 right angles. Other right angles in the classroom include corners of the blackboard, door, books, and floor tiles.

Classifying Angles

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?

Acute means 'sharp' in Latin. An acute angle has a small opening — its arms are close together, forming a sharp pointed shape like a sharp tip.

Obtuse means 'blunt' in Latin. An obtuse angle has a wide, blunt opening — its arms are spread far apart, like a blunt or rounded shape.

The words are chosen because the visual appearance of each angle matches the meaning of the word: acute angles look sharp and pointed, obtuse angles look wide and blunt.

Answer: Acute angles look sharp and pointed (small opening), matching the meaning 'sharp.' Obtuse angles have a wide, blunt opening, matching the meaning 'blunt.'

Where Are the Angles?

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?

(a) A clock face is a full circle (360°) divided into 12 equal hour sections.

Angle between adjacent hour markings = 360° ÷ 12 = 30°.

At 1 o'clock, the hour hand points to 1 and the minute hand points to 12 — they are 1 hour-division apart.

So the angle = 1 × 30° = 30°.

(b) At 2 o'clock: hour hand at 2, minute hand at 12 — 2 divisions apart.

Angle = 2 × 30° = 60°.

At 4 o'clock: 4 divisions apart.

Angle = 4 × 30° = 120°.

At 6 o'clock: 6 divisions apart — the hands point in opposite directions.

Angle = 6 × 30° = 180° (a straight angle).

Answer: (a) The clock face is divided into 12 equal parts, each 30°. At 1 o'clock the hands are 1 division apart, so the angle is 30°. (b) 2 o'clock: 60°; 4 o'clock: 120°; 6 o'clock: 180°.

All 38 questions in Lines and Angles are solved step by step in the Solutions PDF above.

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