NCERT Solutions Ganita Prakash (Part 1) Chapter 7 Two Sides and the Included Angle — Figure it Out

Book page 161 Updated on2026-09-05

Q1.

Construct triangles for the following measurements where the angle is included between the sides: (a) 3 cm, 75°, 7 cm (b) 6 cm, 25°, 3 cm (c) 3 cm, 120°, 8 cm

Answer

In each case draw the longer given side as the base, make the angle at one end, and mark the other side along the new arm.

Set	Base to draw	Angle at that end	Second side along the arm	Shape you get
(a) 3 cm, 75°, 7 cm	7 cm	75°	3 cm	Scalene, acute-angled
(b) 6 cm, 25°, 3 cm	6 cm	25°	3 cm	Scalene, thin
(c) 3 cm, 120°, 8 cm	8 cm	120°	3 cm	Scalene, obtuse-angled

For (a), for example:

Draw PQ = 7 cm.
At P draw an arm making 75° with PQ.
Mark R on that arm with PR = 3 cm.
Join QR. ∆PQR is the required triangle.

Check it yourself: Each set gives one and only one triangle. Two classmates who draw (c) carefully will get triangles that fit exactly on top of each other.

Q2.

We have seen that triangles do not exist for all sets of sidelengths. Is there a combination of measurements in the case of two sides and the included angle where a triangle is not possible? Justify your answer using what you observe during construction.

Answer

Yes — but only when the given angle is not a genuine angle of a triangle, that is when it is 180° or more.

Measurements	What happens on paper	Triangle?
3 cm, 180°, 7 cm	The two arms lie in one straight line; the third vertex falls on the base	No
4 cm, 210°, 6 cm	The angle cannot even be drawn as an interior angle	No
4 cm, 179°, 6 cm	An extremely flat but genuine triangle	Yes

Why it happens: As long as the included angle is more than 0° and less than 180°, the two arms point in different directions, so their far ends can always be joined and a triangle appears — the two sidelengths never matter. Only at 180° do the arms fall into one line and the triangle collapses. So, unlike the three-sides case, here it is the angle alone that decides.

Tip: Compare this with the triangle inequality. There the lengths could fail; here they never do.

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