NCERT Solutions Ganita Prakash (Part 1) Chapter 4 –954.2 Angles in a Quadrilateral — In-text Questions

Book page 94 Updated on2026-09-05

Q1.

Is it possible to construct a quadrilateral with three angles equal to 90° and the fourth angle not equal to 90°?

Answer

No, it is impossible. Try as many constructions as you like — the fourth angle always closes at exactly 90°.

Angle sum of a quadrilateral = 360°
Three right angles use up 90 × 3 = 270°
Fourth angle = 360 – 270 = 90°

There is no room for any other value, so such a quadrilateral cannot exist. (This also proves statement (ii) of the true/false question on page 109: a quadrilateral with three right angles must be a rectangle.)

Tip: this is why the book's shortest definition of a rectangle could have said “three angles are 90°” instead of four. Both tests admit exactly the same figures.

Q2.

But why not?

Answer

Because of a property that holds for every quadrilateral: the sum of all its angles is 360°.

Draw one diagonal. It splits the quadrilateral into two triangles, and each triangle contributes 180°.

180° + 180° = 360°

Once that total is fixed, three angles of 90° leave exactly 90° for the fourth. Nothing about the construction can change it.

Why this is the real reason: failing at ten attempts only tells you the task is hard. The angle-sum property tells you it is impossible — no attempt, however careful, can ever succeed. That is the difference between a conjecture from experiment and a proof.

Q3.

Consider a quadrilateral SOME. Draw a diagonal SM. We get two triangles ∆SEM and ∆SOM. What do we get when we add all six angles?

Answer

We get 360°, and those six angles regroup exactly into the four angles of the quadrilateral.

In ∆SEM: ∠1 + ∠2 + ∠3 = 180°
In ∆SOM: ∠4 + ∠5 + ∠6 = 180°
Adding: ∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 = 180° + 180° = 360°

Regroup: (∠1 + ∠4) + (∠3 + ∠6) + ∠2 + ∠5 = 360°

Here ∠1 + ∠4 is the full angle of the quadrilateral at S, ∠3 + ∠6 is the full angle at M, while ∠2 and ∠5 are the whole angles at E and O. So:

The sum of all angles in any quadrilateral is 360°.

One diagonal turns any quadrilateral into two triangles, so its angle sum is 180° + 180° = 360°.

Why the diagonal has to be inside: for this regrouping to work, the diagonal must lie inside the figure so that ∠1 and ∠4 really are the two parts of the angle at S. In a concave quadrilateral one of the two diagonals falls outside — but the other one still lies inside, so the result survives (see Question 10 of the last Figure it Out).

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