NCERT Solutions for Class 7th Maths Chapter 7 Do triangles always exist? — In-text Questions

Book page 162–163 Updated on2026-09-19

Q1.

Do triangles exist for every combination of two angles and their included side? Explore.

Answer

No. If the two given angles are too big, the two arms never meet.

Two angles	Sum	What the arms do	Triangle?
45°, 80°	125°	Lean towards each other	Yes
90°, 90°	180°	Both go straight up — parallel	No
100°, 95°	195°	Lean away from each other	No

Why it happens: Two arms meet only if together they turn inwards. If both angles are 90° or more, each arm is already vertical or leaning outwards, so they can only drift apart. Just as three lengths must pass a test, two angles must pass one too.

Q2.

Find examples of measurements of two angles with the included side where a triangle is not possible.

Answer

Any pair whose sum reaches 180° will do, whatever the length of the included side.

Measurements	Sum of the two angles	Triangle?
90°, 5 cm, 90°	180°	No — the arms are parallel
110°, 4 cm, 95°	205°	No
40°, 6 cm, 150°	190°	No
120°, 7 cm, 60°	180°	No

Tip: If the two angles are each greater than or equal to a right angle (90°), a triangle is clearly not possible.

Q3.

Now we make one of the base angles an acute angle, say 40°. What are the possible values that the other angle should take so that the lines don’t meet? (a) Try to find a possible ∠B (marked in the figure) for this to happen. (b) What could be smallest value of ∠B for the lines to not meet?

Page 162 — the base AB with the ray l drawn from A at 40°.

Answer

With ∠A = 40°, the arm from B must bend far enough to the right.

(a) ∠B = 150° works — so do 145°, 160°, 170°. With any of these the line from B never meets the line l drawn from A.

(b) The smallest such value is ∠B = 140°.

The line m through B parallel to l makes the smallest ∠B that still avoids l.

Why it happens: The more the arm at B tips to the right, the later it would cross l — until at one special tilt it becomes parallel to l and never crosses at all. That parallel position gives the smallest ∠B that fails, and every larger ∠B fails too.

Q4.

Can you tell the actual value of ∠B be in this case? [Hint: Note that AB is the transversal.]

Answer

∠B = 140°.

Line m through B is parallel to line l through A
AB is the transversal
∠A and ∠B are interior angles on the same side of the transversal
So ∠A + ∠B = 180°
40° + ∠B = 180°
∠B = 140°

Why it happens: This is the co-interior (allied) angle property of parallel lines learnt in the chapter on lines and angles. Because the borderline case is exactly the parallel case, the borderline value of ∠B is exactly 180° minus the other angle.

Q5.

So, for what values of ∠B, does a triangle not exist? Does the length AB play any part here?

Answer

A triangle does not exist when ∠B ≥ 140°, and the length AB plays no part at all.

∠A = 40°
Triangle exists when ∠A + ∠B < 180°, i.e. ∠B < 140°
Triangle fails when ∠B ≥ 140°

In general, for base angles ∠A and ∠B:

∠A + ∠B < 180° → triangle exists
∠A + ∠B ≥ 180° → triangle does not exist

Why it happens: Making AB longer or shorter only slides the arm at B sideways; it does not change the direction the arm points in. Two lines meet or fail to meet purely because of their directions, so only the angles decide.

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