NCERT Solutions Ganita Prakash (Part 1) Chapter 7 Do triangles always exist? — Figure it Out

Book page 163 Updated on2026-09-05

Q1.

For each of the following angles, find another angle for which a triangle is (a) possible, (b) not possible. Find at least two different angles for each category: (a) 30° (b) 70° (c) 54° (d) 144°

Answer

The other angle must be less than 180° minus the given angle for a triangle to be possible.

Given angle	Limit (180° – given)	Triangle possible — two choices (third angle)	Triangle not possible — two choices
(a) 30°	150°	50° (third 100°), 90° (third 60°)	150°, 170°
(b) 70°	110°	60° (third 50°), 80° (third 30°)	120°, 140°
(c) 54°	126°	90° (third 36°), 64° (third 62°)	134°, 154°
(d) 144°	36°	20° (third 16°), 35° (third 1°)	36°, 90°

Sample check, (b) with 60°:
70° + 60° = 130° < 180° ✓
Third angle = 180° – 130° = 50°

Why it happens: The two chosen angles must leave something over for the third angle, so their sum has to stay strictly under 180°. Angles at or above the limit leave nothing (or less than nothing) for the third angle.

Tip: In (d), because 144° is already large, the partner angle has very little room — anything from just above 0° up to just below 36°.

Q2.

Determine which of the following pairs can be the angles of a triangle and which cannot: (a) 35°, 150° (b) 70°, 30° (c) 90°, 85° (d) 50°, 150°

Answer

Add the pair and compare with 180°.

Pair	Sum	Angles of a triangle?	Third angle
(a) 35°, 150°	185°	Cannot	—
(b) 70°, 30°	100°	Can	80°
(c) 90°, 85°	175°	Can	5°
(d) 50°, 150°	200°	Cannot	—

Why it happens: In (c) the sum 175° squeezes past 180° with only 5° to spare, so the triangle is real but very thin and sharp. In (a) and (d) the sum already exceeds 180°, leaving a negative amount for the third angle — impossible.

Q3.

Like the triangle inequality, can you form a rule that describes the two angles for which a triangle is possible? Can the sum of the two angles be used for framing this rule?

Answer

Yes. The rule uses exactly the sum of the two angles.

For two angles ∠A and ∠B of a triangle:
0° < ∠A + ∠B < 180°

If the sum is less than 180°, a triangle with these two angles exists.
If the sum is 180° or more, no such triangle exists.

Rule for lengths	Rule for angles
each length < sum of the other two	sum of two angles < 180°
triangle inequality	angle condition

Why it happens: The third angle is whatever is left over from 180°. A leftover exists only when the first two angles have not already used up the full 180°. This is the first hint of the angle sum property which is proved a page later.

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