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6. Find the value of the unknown angles in the following triangles.
(a) (b) (c) y (d)
110° 150° 125° 50°
x 30° x 60°
x x

Sum of the Lengths of Two Sides of a Triangle (Triangle

Inequality Property)

There are certain conditions based on the lengths of the sides of a triangle that determine whether
a triangle can be formed. These conditions are called inequalities properties of a triangle. Let us
understand it through some activities.

activities
1. Collect some sticks (or strips) of different lengths, say, 8 cm, 9 cm, 10 cm, 11 cm, ..., 20 cm.
Take any three of these sticks and try to form a triangle. Repeat this by choosing different combinations of
three sticks. Suppose you first choose two sticks of lengths 8 cm and 10 cm.
Your third stick has to be of length more than 10 – 8 = 2 cm and less than 8 + 10 = 18 cm.
Are you able to form a triangle by using a third stick of length either of more than 2 cm or less than 18 cm?
To form a triangle, you will need any three sticks such that the sum of the lengths of any two of them will
always be greater than the length of the third stick. This also suggests that the sum of the lengths of any two
sides of a triangle is greater than the third side.
Let us take some sticks of different lengths and try to form a triangle.
8 cm 9 cm 8 cm 9 cm 8 cm 9 cm

18 cm 17 cm 16 cm
8 cm + 9 cm < 18 cm 8 cm + 9 cm = 17 cm 8 cm + 9 cm > 16 cm
2. Draw any two triangles, say DABC and DPQR in your notebook.
Measure the lengths of their sides by using a ruler and then tabulate your results as follows.
Name of D Lengths of Sides Inequality Is this true?
AB______ AB – BC < CA ____ – _____ < _____ (Yes/No)
AB + BC > CA ____ + _____ > _____ (Yes/No)
BC______ BC – CA < AB ____ – _____ < _____ (Yes/No)
DABC
BC + CA > AB ____ + _____ > _____ (Yes/No)
CA______ CA – AB < BC ____ – _____ < _____ (Yes/No)
CA + AB > BC ____ + _____ > _____ (Yes/No)
PQ_____ PQ – QR < RP ____ – _____ < _____ (Yes/No)
PQ + QR > RP ____ + _____ > _____ (Yes/No)
QR_____ QR – RP < PQ ____ – _____ < _____ (Yes/No)
DPQR
QR + RP > PQ ____ + _____ > _____ (Yes/No)
RP_____ RP – PQ < QR ____ – _____ < _____ (Yes/No)
RP + PQ > QR ____ + _____ > _____ (Yes/No)

We can observe from the above activity that a triangle can be formed only when the sum of the
lengths of any two sides is always greater than the third side.

179 The Triangle and Its Properties