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State true or false.

The difference of any two sides of a triangle is less than the third side.

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True

False

answer is A.

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Detailed Solution

Consider a triangle

A B C

Consider

A C > A B

.
Then, along

A C

constructing

A D = A B

and joining

B D

.
In

Δ A B D

,
Since

A D = A B

∠ A B D = ∠ A D B

..….(1)
Since angles opposite to equal sides

∠ A B D = ∠ 2

……(2)
Evaluate the relationship between the angles.
Therefore,

∠ 1 < ∠ A B D

∠ 1 = ∠ 2

……(3)
Now, from (2) since,

∠ 2

is an exterior angle,

∠ 2 > ∠ 3

……(4)
Evaluate Using (3) and (4),

∠ 1 > ∠ 3

Since the side

B C

opposite to a greater angle is longer,

B C > D C

B C > A C − A D

Since,

A D = A B

B C > A C − A B

Hence the difference of

2

sides of the triangle is less than the third side.
Hence the statement is true.
Hence option 1 is correct.

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