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NCERT Solutions Ganita Prakash Chapter 2 Figure it Out — Types of Angles and their Measures

Book page 53 & 54 Updated on2026-09-05

Q1.
Draw angles with the following degree measures: a. 140° b. 82° c. 195° d. 70° e. 35°
Answer

Angles up to 180° are drawn directly with a protractor. For 195° a small extra step is needed.

AngleHow to draw itType
a. 140°base ray on 0, mark the 140 point, joinobtuse
b. 82°mark the 82 point (just short of 90)acute
c. 195°it is more than 180°, so draw 360° − 195° = 165° first, then mark the curve on the outer sidereflex
d. 70°mark the 70 pointacute
e. 35°mark the 35 pointacute
Drawing a reflex angle: a protractor stops at 180°, so we draw the other angle of the pair and then show, with a curve going the long way round, that we mean the reflex one. Another way for 195° is 180° + 15° — draw a straight angle and turn 15° more.
Q2.
Estimate the size of each angle and then measure it with a protractor: a, b, c, d, e, f. Classify these angles as acute, right, obtuse or reflex angles.
Answer

First estimate, then measure — the aim is to make your guesses better each time. Comparing with the landmarks you already know is the quickest way to estimate:

square corner = 90°    half of it = 45°    straight line = 180°    full turn = 360°
a. 45°acuteb. 166°obtusec. 120°obtused. 33°acutee. 98°obtusef. 348°reflex
The six angles of page 53, measured with a protractor.
AngleMeasureTypeHow to spot it
a.45°acuteexactly half a square corner
b.166°obtusealmost a straight line, but not quite
c.120°obtusea third of a full turn
d.33°acutea little more than a third of a right angle
e.98°obtusejust past a square corner
f.348°reflexalmost a full turn — the two arms nearly meet

For the reflex angle (f) the protractor cannot be used directly: measure the small angle between the arms (about 12°) and subtract it from a full turn.

360° − 12° = 348°
Tip: a good habit is to say to yourself “a little less than a square corner” or “a bit more than a straight line” before touching the protractor. Readings within a degree or two of the table above are correct.
Q3.
Make any figure with three acute angles, one right angle and two obtuse angles.
Answer
50°A65°B80°C90°D115°E140°F
An open zig-zag figure with three acute angles (50°, 65°, 80°), one right angle (90°) and two obtuse angles (115°, 140°).

Draw a zig-zag path of seven line segments and mark the angle at each of its six corners:

CornerAngleType
A50°acute
B65°acute
C80°acute
D90°right
E115°obtuse
F140°obtuse

How to draw it: draw the first segment, use the protractor to make an angle of 50° at its end and draw the next segment, then 65° at the next corner, and so on. The figure need not close up.

Why an open figure is used: a closed six-sided figure cannot have exactly these angles. The six angles of a hexagon must add up to 720°, but three acute angles (each below 90°), one right angle and two obtuse angles (each below 180°) can add up to at most 90 + 90 + 90 + 90 + 180 + 180 = 720°, and in fact always to less than 720°. So the figure has to be open, or made of two separate shapes.
Try This: draw a triangle with three acute angles (say 60°, 70°, 50°) and, next to it, a quadrilateral containing one right angle and two obtuse angles. Together the two shapes also answer the question.
Q4.
Draw the letter 'M' such that the angles on the sides are 40° each and the angle in the middle is 60°.
Answer
40°40°60°
The letter M with side angles of 40° each and a middle angle of 60°.

Steps:

  1. Draw the two outer strokes of the M, slanting slightly outwards at the bottom.
  2. At the top of the left stroke, use the protractor to turn 40° and draw the first slanting stroke going down to the middle.
  3. At the middle point, turn through 60° and draw the second slanting stroke going up.
  4. At the top of the right stroke, the angle should again measure 40°. Check it with the protractor.
Left side angle = 40°    Middle angle = 60°    Right side angle = 40°
Tip: the middle of the M is a sharp V. Since 60° is smaller than the 80° you would get from two vertical strokes, the outer strokes must lean out a little at the bottom. Draw lightly in pencil first, measure, and then darken the lines.
Q5.
Draw the letter 'Y' such that the three angles formed are 150°, 60° and 150°.
Answer
60°150°150°
The letter Y with angles of 150°, 60° and 150° around the joint. They add up to 360°.

Steps:

  1. Mark a point O — this will be the joint of the Y — and draw the stem straight down from it.
  2. From the stem, turn 150° on one side and draw the first upper arm.
  3. From that arm, turn 60° and draw the second upper arm.
  4. The angle between the second arm and the stem is then automatically 150°. Check it.
150° + 60° + 150° = 360° ✔
Why the three angles must add to 360°: the three arms all start at the same point O, and together the three angles go once completely around that point — and one complete turn is 360°.
Q6.
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?
Answer
15°
The Ashoka Chakra: 24 equally spaced spokes, so the angle between two neighbouring spokes is 360° ÷ 24 = 15°.

Angle between two neighbouring spokes: the 24 spokes divide the full turn into 24 equal parts.

360° ÷ 24 = 15°

Largest acute angle between two spokes: every angle between two spokes is a multiple of 15°, and an acute angle must be less than 90°.

15°, 30°, 45°, 60°, 75°, 90° …
The largest multiple of 15 that is still below 90 is 75°

Answer: 15° between neighbouring spokes, and 75° is the largest acute angle.

Did you know? The Ashoka Chakra on our national flag is taken from the Lion Capital of Ashoka at Sarnath. Its 24 spokes are equally spaced, which is why every angle in it is a whole number of degrees.
Q7.
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?
Answer

Let the angle be x. Write down what the puzzle demands.

4x < 90°  →  x < 22.5°  (so that 4 times it is still acute)
5x > 90°  →  x > 18°  (so that 5 times it is obtuse)

So the measure must lie between 18° and 22.5°:

18° < x < 22.5°

Taking whole numbers of degrees, the possibilities are

19°,   20°,   21°,   22°
x2x3x4x5x
19°38°57°76° (acute ✔)95° (obtuse ✔)
20°40°60°80° (acute ✔)100° (obtuse ✔)
21°42°63°84° (acute ✔)105° (obtuse ✔)
22°44°66°88° (acute ✔)110° (obtuse ✔)
Why only these: if x were 18° or less, then 5x would be 90° or less — not obtuse. If x were 22.5° or more, then 4x would be 90° or more — not acute. Notice that the conditions on 2x and 3x are automatically satisfied once 4x is acute.
Did you know? If fractions of a degree are allowed, every value between 18° and 22.5° works — for example 18.5° or 22.4°. The four whole-number answers are the ones expected here.
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