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 : **

In a right triangle, apart from the right angle, the other two angles are x+1 and 2x+5. find the angles of the triangle.

**Solution :**

We know that,

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

90 + (x + 1) + (2x + 5) = 180°

3x + 6 = 90°

3x = 84°

x = 28°

So,

x + 1 = 28 + 1 = 29°

2x + 5 = 2(28) + 5 = 56 + 5 = 61°

So, the three angles of a triangle are 90°, 29° and 61°.

**Problem 2 : **

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

**Solution :**

Let "x" be the first angle.

The second angle = 3(x+3)

The third angle = 2x + 3

We know that,

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

x + 3(x+3) + 2x + 3 = 180°

x + 3x + 9 + 2x + 3 = 180°

6x + 12 = 180°

6x = 168°

x = 28°

The first angle = 28°

The second angle = 3(28+3) = 93°

The third angle = 2(28) + 3 = 59°

So, the three angles of a triangle are 28°, 93° and 59°.

**Problem 3 : **

In a triangle, the ratio between the first and second angle is 1 : 2 and the third angle is 72. Find the first and second angle of the triangle. ** **

**Solution :**

From the given ratio,

The first angle = x

The second angle = 2x

We know that,

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

x + 2x + 72 = 180°

3x = 108°

x = 36°

The first angle = 36°

The second angle = 2(36°) = 72°

So, the first angle is 36° and the second angle is 72°.

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