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 : **

If the angles of a triangle are in the ratio 2 : 7 : 11, then find the angles.

**Solution :**

From the ratio 2 : 7 : 11,

the three angles are 2x, 7x, 11x

In any triangle, sum of the angles = 180°

So, 2x + 7x + 11x = 180°

20x = 180 -------> x = 9

Then, the first angle = 2x = 2(9) = 18°

The second angle = 7x = 7(9) = 63°

The third angle = 11x = 11(9) 99°

Hence the angles of the triangle are (18°, 63°, 99°)

**Problem 2 : **

In a triangle, if the second angle is 10% more than the first angle and the third angle is 20% less than the first angle, then find the three angles of the triangle.

**Solution :**

Let "x" be the first angle.

The second angle = 120 % of x = 1.2x

The third angle = 80% of x = 0.8x

We know that,

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

x + 1.2x + 0.8x = 180°

3x = 180°

x = 60°

The first angle = 60°

The second angle = 1.2(60) = 72°

The third angle = 0.8(60) = 48°

So, the three angles of a triangle are 60°, 72° and 48°.

**Problem 3 : **

If 3 consecutive positive integers be the angles of a triangle, then find the three angles of the triangle.

**Solution :**

Let "x" be the first angle.

The second angle = x + 1

The third angle = x + 1 + 1 = x + 2

We know that,

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

x + x + 1 + x + 2 = 180°

3x + 3 = 180°

3x = 177°

x = 59°

The first angle = 59°

The second angle = 59 + 1 = 60°

The third angle = 60 + 1 = 61°

So, the three angles of a triangle are 59°, 60° and 61°.

**Problem 4 : **

In a triangle, if the second angle is 2 times the first angle and the third angle is 3 times the first angle, find the angles of the triangle.

**Solution :**

Let "x" be the first angle.

The second angle = 2x

The third angle = 3x

We know that,

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

x + 2x + 3x = 180°

6x = 180°

x = 30°

The first angle = 30°

The second angle = 2(30°) = 60°

The third angle = 3(30°) = 90°

So, the three angles of a triangle are 30°, 60° and 90°.

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