NCERT Solutions Ganita Prakash Chapter 2 In-text Questions — The Labelled Protractor

Book page 36 & 37 Updated on2026-09-05

Q1.

There are two sets of numbers on the protractor: one increasing from right to left and the other increasing from left to right. Why does it include two sets of numbers?

Answer

So that the protractor can be used from either side without any subtraction.

If the base arm of your angle points to the right, start at the 0 on the right and read the inner set of numbers.
If the base arm points to the left, start at the 0 on the left and read the outer set.

Inner reading + outer reading = 180° at every mark
(for example 20 and 160, 35 and 145, 90 and 90)

Why it helps: without the second set you would have to turn the paper around, or measure one number and subtract it from 180 every time. The two scales save that step.

Tip: always check first which arm lies on the 0 mark, then follow that same scale to the other arm. Mixing the two scales is the most common protractor mistake.

Q2.

Name the different angles in the figure and write their measures. Did you include angles such as ∠TOQ? Which set of markings did you use — inner or outer?

Answer

The six rays read on one scale: P at 0°, Q at 35°, R at 95°, S at 125°, T at 160° and U at 180°.

The rays OP, OQ, OR, OS, OT and OU start from the centre O. Reading from the 0 mark at P (that is, using one single scale throughout):

P → 0° Q → 35° R → 95° S → 125° T → 160° U → 180°

Every angle is now just a subtraction of two readings:

Angle	Working	Measure
∠POQ	35 − 0	35°
∠POR	95 − 0	95°
∠POS	125 − 0	125°
∠POT	160 − 0	160°
∠POU	180 − 0	180° (straight angle)
∠QOR	95 − 35	60°
∠QOS	125 − 35	90°
∠QOT	160 − 35	125°
∠QOU	180 − 35	145°
∠ROS	125 − 95	30°
∠ROT	160 − 95	65°
∠ROU	180 − 95	85°
∠SOT	160 − 125	35°
∠SOU	180 − 125	55°
∠TOU	180 − 160	20°

Yes, angles such as ∠TOQ must also be included — it is the same as ∠QOT = 125°. In all, 15 angles can be named from these six rays.

Tip: use one set of markings — inner or outer — for the whole figure. Here the outer scale is convenient because OT and OS fall on the round numbers 20 and 55 on it.

Q3.

What is the measure of ∠TOS? Can you use the numbers marked to find the angle without counting the number of markings? Here, OT and OS pass through the numbers 20 and 55 on the outer scale. How many units of 1 degree are contained between these two arms? Can subtraction be used here?

Answer

Yes — subtraction is exactly the short cut we need.

OS is at 55 OT is at 20 (outer scale)
∠TOS = 55 − 20 = 35°

So there are 35 units of 1° between the two arms, and we did not have to count a single mark.

Why subtraction works: the reading of an arm tells us how far that arm has turned from the 0 mark. If one arm has turned 20 units and the other 55 units from the same starting line, the turn between them is the difference, 55 − 20.

Check it yourself: read the same two arms on the inner scale — they are at 160 and 125, and 160 − 125 = 35°. The same answer, as it must be.

Q4.

How can we measure angles directly without having to subtract? Place the protractor so the centre is on the vertex of the angle. Align the protractor so that one of the arms passes through the 0º mark. What is the degree measure of ∠AOB?

Answer

If one arm is placed on the 0 mark, then the reading of the other arm is the answer — the subtraction becomes 0 and disappears.

Reading of OB = 0
Reading of OA = 80
∠AOB = 80 − 0 = 80°

The three steps to remember:

Centre on vertex — put the mid-point of the protractor's base exactly on O.
Base on one arm — turn the protractor until OB lies along the 0 line.
Read the other arm — follow the scale that starts from 0 on OB up to OA.

Check it yourself: measured on the printed figure the ray OA stands at 80.3° above OB, so 80° is the reading the picture is meant to give. If an arm is too short to reach the scale, extend it with a ruler first — making an arm longer never changes the angle.

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