**Properties of Triangle 1 :**

This is the continuity of our web content given on "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°.

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.

2. 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

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.

Hence, 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°

Since 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°

Since 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.

The second angle = x + 5

The third angle = x + 5 + 5 = x + 10

We know that,

the sum of the three angles of a triangle = 180°

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°

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

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