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°.

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°)

**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}

x^{2} - 24x + 144 = x^{2} - 26x + 169 + x^{2} - 28x + 196

x^{2} - 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°.

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