NCERT Solutions Ganita Prakash (Part 1) Chapter 7 Finding the third angle — Figure it Out

Book page 165 Updated on2026-09-05

Q1.

Find the third angle of a triangle (using a parallel line) when two of the angles are: (a) 36°, 72° (b) 150°, 15° (c) 90°, 30° (d) 75°, 45°

Answer

Draw XY through A parallel to BC. The two base angles reappear at A as alternate angles, so the three angles at A fill a straight angle.

∠XAB + ∠BAC + ∠YAC = 180°
∠B + ∠BAC + ∠C = 180°
Third angle = 180° – (sum of the other two)

Two angles	Sum	Third angle
(a) 36°, 72°	108°	72°
(b) 150°, 15°	165°	15°
(c) 90°, 30°	120°	60°
(d) 75°, 45°	120°	60°

Worked out fully for (a), as the book does:

∠XAB = ∠B = 36° and ∠YAC = ∠C = 72° (alternate angles)
∠XAB + ∠BAC + ∠YAC = 180°
36° + ∠BAC + 72° = 180°
108° + ∠BAC = 180°
∠BAC = 72°

Tip: (a) and (b) each give an isosceles triangle, because the third angle equals one of the given ones.

Q2.

Can you construct a triangle all of whose angles are equal to 70°? If two of the angles are 70° what would the third angle be? If all the angles in a triangle have to be equal, then what must its measure be? Explore and find out.

Answer

No, such a triangle is impossible.

70° + 70° + 70° = 210°
210° ≠ 180° → no such triangle

If two of the angles are 70°:

Third angle = 180° – (70° + 70°)
= 180° – 140°
= 40°

If all three angles are equal:

Let each angle be x
x + x + x = 180°
3x = 180°
x = 60°

Why it happens: The three angles have a fixed budget of 180° to share. Three angles of 70° overspend it by 30°. Sharing the budget equally gives 180° ÷ 3 = 60° each — which is exactly why every equilateral triangle has three 60° angles.

Q3.

Here is a triangle in which we know ∠B = ∠C and ∠A = 50°. Can you find ∠B and ∠C?

Answer

∠B = ∠C = 65°.

An isosceles triangle with apex angle 50°; the two equal base angles come out as 65° each.

∠A + ∠B + ∠C = 180°
50° + ∠B + ∠C = 180°
∠B + ∠C = 130°
But ∠B = ∠C, so 2 × ∠B = 130°
∠B = 130° ÷ 2 = 65°
∠C = 65°

Why it happens: The angle sum property tells us how much the two base angles share between them; the condition ∠B = ∠C tells us to split that share equally. Two facts, two unknowns, one clean answer.

Check it yourself: 50° + 65° + 65° = 180° ✓

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