NCERT Solutions Ganita Prakash (Part 1) Chapter 7 Finding the third angle — In-text Questions
Book page 164 Updated on2026-09-05
Q1.
Let us take two angles, say 60° and 70°, whose sum is less than 180°. Let the included side be 5 cm. What could the measure of the third angle be? Does this measure change if the base length is changed to some other value, say 7 cm? Construct and find out.
Answer
The third angle is 50°, and it does not change when the base changes.
Why it happens: Changing the base only makes the triangle bigger or smaller; the arms keep the same directions, so the corner they form keeps the same opening. Size changes, shape does not.
Q2.
In general, once the two angles are fixed, does the third angle depend on the included sidelength? Try with different pairs of angles and lengths.
Answer
No. The third angle depends only on the two given angles.
Two angles
Included side
Third angle
45°, 80°
3 cm
55°
45°, 80°
8 cm
55°
30°, 30°
4 cm
120°
30°, 30°
9 cm
120°
In every row: third angle = 180° – (sum of the two given angles)
Tip: If your measurement differs by a degree or two, it is drawing error, not mathematics. Sharpen the pencil and read the protractor from directly above.
Q3.
Try experimenting with different triangles to see if there is a relation between any two angles and the third one. To find this relation, what data will you keep track of and how will you organise the data you collect?
Answer
Keep a three-column table of the three angles, and add a fourth column for their sum.
∠A
∠B
∠C
∠A + ∠B + ∠C
60°
70°
50°
180°
45°
80°
55°
180°
90°
30°
60°
180°
120°
30°
30°
180°
60°
60°
60°
180°
The relation:
third angle = 180° – (sum of the other two) equivalently, ∠A + ∠B + ∠C = 180°
Why it happens: Recording only two angles at a time hides the pattern; recording the sum makes it jump out, because the last column is the same number every time. Choosing what to tabulate is half the discovery.
Check it yourself: Draw five very different triangles — thin, fat, right-angled, obtuse — and fill the table. The last column stays 180°.
Q4.
Consider a triangle ABC with ∠B = 50° and ∠C = 70°. Let us suppose we construct a line XY parallel to BC through vertex A. We can see new angles being formed here: ∠XAB, and ∠YAC. What are their values? Can we find ∠BAC from this?
Answer
∠XAB = 50°, ∠YAC = 70°, and therefore ∠BAC = 60°.
XY is parallel to BC, so the two outer angles at A copy ∠B and ∠C.
XY ∥ BC, with AB as a transversal → ∠XAB = ∠B = 50° (alternate angles) XY ∥ BC, with AC as a transversal → ∠YAC = ∠C = 70° (alternate angles)
∠XAB + ∠BAC + ∠YAC = 180° (they form a straight angle at A) 50° + ∠BAC + 70° = 180° 120° + ∠BAC = 180° Thus ∠BAC = 60°
Why it happens: The parallel line carries the two base angles up to the vertex A, where they lie side by side with ∠BAC along a straight line. A straight angle is 180°, so the three of them must add to 180° — and that is exactly the angle sum property.