# PROPERTIES OF TRIANGLE

The following are the two important properties of triangle.

1. The sum of the lengths of any two sides of a triangle is greater than the third side.

2. The sum of all the three angles of a triangle is 180°.

## Some Other Properties of Triangle

1.  In an equilateral triangle, all the three sides and three angles will be equal and each angle will measure 60°.

2. In an isosceles triangle, the lengths of two of the sides will be equal. And the corresponding angles of the equal sides will be equal.

3. In a right triangle, square of the hypotenuse is equal to the sum of the squares of other two sides. This is known as Pythagorean theorem.

Hypotenuse :

The longest side of in the right triangle which is opposite to  right angle (90°)

## Practice Problems

Problem 1 :

The sides of an equilateral triangle are shortened by 12 units, 13 units and 14 units respectively and a right angle triangle is formed. Find the length of each side of the equilateral triangle.

Solution :

Let 'x' be the length of each side of the equilateral triangle.

Then, the sides of the right angle triangle are

(x - 12), (x - 13) and (x - 14)

In the above three sides, the side represented by (x -12) is hypotenuse (because that is the longest side).

Using Pythagorean theorem,

(x - 12)2  =  (x - 13)2 + (x - 14)2

x2 - 24x + 144  =  x2 - 26x + 169 + x2 - 28x + 196

x2 - 30x + 221  =  0

(x - 13)(x - 17)  =  0

x  =  13  or  x  =  17

x  =  13 can not be accepted. Because, if x  =  13, one of the sides of the right angle triangle would be negative.

So, the side of the equilateral triangle is 17 units.

Problem 2 :

Can 30°, 60° and 90° be the angles of a triangle ?

Solution :

Let us add all the three given angles and check whether the sum is equal to 180°.

30° +  60° + 90°  =  180°

Because the sum of the angles is equal 180°, the given three angles can be the angles of a triangle.

Problem 3 :

Can 35°, 55° and 95° be the angles of a triangle ?

Solution :

Let us add all the three given angles and check whether the sum is equal to 180°.

35° +  55° + 95°  =  185°

Because the sum of the angles is not equal 180°, the given three angles can not be the angles of a triangle.

Problem 4 :

In a triangle, If the second angle is 5° greater than the first angle and the third angle is 5° greater than second angle, find the three angles of the triangle.

Solution :

Let 'x' be the first angle.

Then,

Second angle  =  x + 5

Third angle  =  x + 5 + 5  =  x + 10

We know that, the sum of the three angles of a triangle is 180°.

Then,

x + (x + 5) + (x + 10)  =  180°

3x + 15  =  180

3x  =  165

x  =  55

The first angle  =  55°

The second angle  =  55 + 5  =  60°

The third angle  =  60 + 5  =  65°

So, the three angles of a triangle are 55°, 60° and 65°.

Apart from the problems given above, if you need more problems on triangle properties, please click the following links.

Triangle Properties

Triangle Properties 2

Triangle Properties 3

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