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NCERT Solutions Ganita Prakash Chapter 2 – 31Section 2.8 Right Angles and Perpendicular Lines — Figure it Out

Book page 29 Updated on2026-09-05

Q1.
How many right angles do the windows of your classroom contain? Do you see other right angles in your classroom?
Answer

A window is usually a rectangle, and every rectangle has 4 right angles — one at each corner.

1 window pane = 4 right angles
A window with 4 panes = 4 × 4 = 16 right angles
(plus the 4 of the outer frame)

Count the panes in your own window and multiply by 4.

Other right angles in the classroom:

  • corners of the blackboard, the door, the notebook, the desk top and the floor tiles;
  • the corner where two walls meet, and where a wall meets the floor;
  • the corner of a chart paper, a duster, a cupboard and the almirah shelves;
  • the two edges of a set-square, and the corner of your geometry box.
Check it yourself: fold a paper twice to make a right-angle tester, and hold it against these corners.
Q2.
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?
Answer

A straight angle at A means the two arms must go in exactly opposite directions from A — so the point you join must lie on the ray AB extended backwards through A.

AB180°AB180°
A straight angle at A: join A to a grid point on the ray opposite to AB — straight left in the first grid, and up-left along the diagonal in the second.
GridStep from A to BWhere to join AHow many such grid points
First (AB horizontal)3 steps to the rightany grid point of the same row to the left of A3
Second (AB slanting)2 steps right and 2 steps downup-left along the same slant — 1 left & 1 up, or 2 left & 2 up2

In every case the angle formed is 180°.

How many different ways? Only one direction works in each grid — the direction exactly opposite to AB. All the grid points lying that way give the same straight angle, because they all lie on one and the same ray. So there are several points to choose from, but only one straight angle.
Q3.
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.)
Answer

For a right angle at A the new line must be perpendicular to AB. On a square grid there is a neat trick for this.

If B is reached from A by a steps right and b steps down,
then a perpendicular direction is b steps right and a steps up
(or b steps left and a steps down)
AB90°AB90°
A right angle at A: in the first grid go straight up or straight down; in the second go 2 left and 2 down, or 2 right and 2 up.
GridStep from A to BPerpendicular steps from ADifferent ways
First (AB horizontal)3 rightstraight up (2 points) or straight down (3 points)2
Second (AB slanting)2 right, 2 down2 right & 2 up, or 2 left & 2 down2

So in each grid there are two different right angles at A — one on each side of the line AB — and each may be drawn to any grid point along that direction.

Check it yourself: use the hint — extend BA beyond A to C. The line you draw must cut the straight angle ∠CAB into two equal halves, so each half is 180° ÷ 2 = 90°.
Q4.
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.
Answer
4 right angles
Two creases crossing at right angles — four exact right angles of 90° each.

a. There are 4 right angles — the two creases cross each other and cut the whole turn around that point into four equal parts.

Full turn = 360°
4 equal angles → each = 360° ÷ 4 = 90°

Justification: the second fold was made by bringing one half of the first crease exactly onto the other half. Superimposition therefore makes the two angles on one side equal, and together they form a straight angle:

two equal angles adding to 180° → each = 90°

The same argument works on the other side of the crease, giving 4 exact right angles.

b. Instructions anyone can follow:

  1. Take a sheet of paper and fold it once in any slanting direction. Press the fold hard and open it — you get a slanting crease.
  2. Now fold the paper a second time, this time bringing one part of the first crease exactly on top of the other part of the same crease.
  3. Press the second fold and open the paper. The two creases meet at a point and form four right angles.
Tip: this is the quickest way to make a perfect 90° without a protractor — the paper does the measuring for you.
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