NCERT Solutions Ganita Prakash Chapter 2 – 43Section 2.9 Measuring Angles with a Protractor — Figure it Out
Book page 40 Updated on2026-09-05
Q1.
Find the degree measures of the following angles using your protractor.
Answer
Place the centre of the protractor on the vertex H, put one arm on the 0 line, and read the other arm on the same scale.
The three angles of page 40 — 47°, 23° and 108°, drawn in different directions.
Figure
Angle
Measure
Type
1st (vertex H at the bottom)
∠IHJ
47°
acute
2nd (vertex H at the top)
∠GHK
23°
acute
3rd (vertex H on the left)
∠IHJ
108°
obtuse
In the middle figure I and J lie on the very same two rays as G and K, so that angle may equally well be named ∠IHJ.
Tip: if an arm is too short to reach the curved scale, extend it with your ruler. The angle does not change when the arms are made longer. (Measured exactly, the three angles are 47.0°, 23.5° and 108.8°.)
Q2.
Find the degree measures of different angles in your classroom using your protractor.
Answer
This is an activity — measure and record. Here is a table you can copy and fill in:
Object
Where the angle is
Expected measure
Notebook / blackboard corner
between two edges
90°
Open door
between the door and the wall
anything from 0° to about 180°
Open scissors
between the blades
an acute angle, say 30°–60°
Hands of the wall clock
at the centre
a multiple of 30° at exact hours
Set-square
its three corners
90°, 60°, 30° (or 90°, 45°, 45°)
Open geometry box lid
between lid and box
an obtuse angle
Check it yourself: measure the three corners of your set-square and add them. You should get 180° — the same result as question 9 below.
Q3.
Find the degree measures for the angles given below. Check if your paper protractor can be used here!
Answer
First angle: ∠IHJ = 42° (acute) Second angle: ∠IHJ = 116° (obtuse)
(Measured exactly on the printed page they are 41.7° and 116.3°.)
No, the paper protractor cannot be used here.
Why not: the handmade paper protractor was built only by repeated halving, so it carries just the marks 0°, 22.5°, 45°, 67.5°, 90°, 112.5°, 135°, 157.5° and 180°. The angles 42° and 116° do not fall on any of those creases. A standard protractor is needed because it is divided into all 180 single degrees.
Tip: the paper protractor can still tell you a lot — 42° lies between 22.5° and 45° (so it is acute), and 116° lies between 112.5° and 135° (so it is obtuse).
Q4.
How can you find the degree measure of the angle given below using a protractor?
Answer
The marked angle is a reflex angle, bigger than 180°, so it cannot be read directly — a protractor only goes up to 180°.
Method: measure the other angle at the same vertex (the unmarked one), then subtract it from a full turn.
Why it works: the marked angle and the unmarked angle together make one complete turn about the vertex, and a complete turn is 360°.
Note: the unmarked angle of the printed figure measures 101.5°, which gives a reflex angle of about 258°. The textbook's answer key rounds the reading to 100° and so gives 260°. Any answer between about 258° and 260° is right — what matters is the method.
Another way: split the reflex angle into a straight angle plus the rest — 180° + 78° ≈ 258° — and add the two parts.
Q5.
Measure and write the degree measures for each of the following angles: a, b, c, d, e, f.
Answer
Place the centre of the protractor on each vertex, one arm on 0, and read off the other arm:
Angle
Measure
Type
a.
80°
acute
b.
120°
obtuse
c.
60°
acute
d.
130°
obtuse
e.
130°
obtuse
f.
60°
acute
Check it yourself: measured very precisely on the printed page the six angles come to 79°, 120.5°, 59.5°, 129°, 128° and 61°. So a reading within a degree or two of the values above is perfectly correct.
Tip: angles (d) and (e) are equal even though they are drawn in different directions and with arms of different lengths — turn your protractor, not your judgement. The same is true of (c) and (f).
Q6.
Find the degree measures of ∠BXE, ∠CXE, ∠AXB and ∠BXC.
Answer
A, X and E are in a straight line. Reading from E: C at 85°, B at 115°, A at 180°.
A, X and E lie in one straight line, so ∠AXE = 180°. Reading the rays from the 0 mark at E:
E → 0° C → 85° B → 115° A → 180°
Angle
Working
Measure
∠BXE
115 − 0
115°
∠CXE
85 − 0
85°
∠AXB
180 − 115
65°
∠BXC
115 − 85
30°
Check it yourself: ∠AXB + ∠BXE = 65° + 115° = 180° ✔ — as it must be, because AXE is a straight line.
Q7.
Find the degree measures of ∠PQR, ∠PQS and ∠PQT.
Answer
Reading from the arm QP: R at 45°, S at 100° and T at 150°.
The vertex is Q. Keeping QP on the 0 mark and reading the same scale for the other three rays:
Angle
Reading
Measure
Type
∠PQR
45 − 0
45°
acute
∠PQS
100 − 0
100°
obtuse
∠PQT
150 − 0
150°
obtuse
Tip: from the same readings you also get ∠RQS = 100 − 45 = 55°, ∠SQT = 150 − 100 = 50° and ∠RQT = 150 − 45 = 105°.
Q8.
Make the paper craft as per the given instructions. Then, unfold and open the paper fully. Draw lines on the creases made and measure the angles formed.
Answer
This is a folding activity. Do it step by step and then measure.
Fold the square sheet as shown in pictures 1 to 8, pressing every crease firmly.
Open the paper out fully. You will see a pattern of creases meeting at the centre.
Draw a line along every crease with a pencil and ruler.
Measure each angle at the centre with your protractor.
What you should find: because each fold halves the previous one, the angles at the centre come out as neat fractions of a full turn:
360° ÷ 4 = 90° 360° ÷ 8 = 45° 360° ÷ 16 = 22.5°
Check it yourself: all the angles around the centre point must add up to 360°. Add your readings — if they do not total 360°, measure again.
Q9.
Measure all three angles of the triangle shown in Fig. 2.21 (a), and write the measures down near the respective angles. Now add up the three measures. What do you get? Do the same for the triangles in Fig. 2.21 (b) and (c). Try it for other triangles as well, and then make a conjecture for what happens in general!
Answer
Measure each corner of the triangle and add. For Fig. 2.21(a):
Fig. 2.21(a) measured with a protractor: 48° + 68° + 64° = 180°.
∠A + ∠B + ∠C = 48° + 68° + 64° = 180°
Doing the same for the other two triangles of Fig. 2.21:
Triangle
∠A
∠B
∠C
Sum
(a)
48°
68°
64°
180°
(b)
58°
63°
59°
180°
(c)
31°
52°
97°
180°
The conjecture: the three angles of any triangle always add up to 180° — a straight angle — whatever its shape or size.
A picture that hints at why: tear off the three corners of a paper triangle and place them side by side with their vertices touching. They fit together perfectly along a straight line. You will learn the full proof in a later class.
Tip: if your three readings add up to 179° or 181°, that is only a small measuring error. Measure once more, keeping the centre of the protractor exactly on the vertex.