**Problem 1 : **

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

**Problem 2 : **

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

**Problem 3 : **

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.

**Problem 4 : **

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

**Problem 5 : **

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

**Problem 6 : **

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

**Problem 7 : **

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.

**Problem 8 : **

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.

**Problem 9 : **

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.

**Problem 10 : **

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.

**Problem 11 : **

The first angle of a triangle is two-third of the third angle, the second angle is double the first angle. Find the three angles of the triangle.

**Problem 12 : **

In a triangle, sum of the first and second angles is 95°, sum of the second and third angles is 130° and sum of the first and third angles is 135°. Find the three angles of the triangle.

**Problem 1 : **

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 2 : **

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 3 : **

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, 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°

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

**Problem 4 : **

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

**Solution :**

The angles of the triangle are in the ratio 2 : 7 : 11.

Then, the three angles are

2x, 7x and 11x

In any triangle,

Sum of the three angles = 180°

So, we have

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°

So, the angles of the triangle are 18°, 63° and 99°.

**Problem 5 : **

In a triangle, if the second angle is 20% 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.

Then, 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 6 : **

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.

Then, the second angle = x + 1

The third angle = 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 = 59 + 2 = 61°

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

**Problem 7 : **

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.

Then 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°.

**Problem 8 : **

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, we have

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 9 : **

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.

Then, 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 10 : **

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 :**

The ratio of the first angle and second angle is 1 : 2.

Then, 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°.

**Problem 11 : **

The first angle of a triangle is two-third of the third angle, the second angle is double the first angle. Find the three angles of the triangle.

**Solution :**

Let x be the third angle.

Then, the first angle is

= 2x/3

Second angle is

= 2 ⋅ 2x/3

= 4x/3

We know that,

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

Then,

x + 2x/3 + 4x/3 = 180°

3x/3 + 2x/3 + 4x/3 = 180°

(3x + 2x + 4x) / 3 = 180°

9x / 3 = 180°

3x = 180°

Divide each side by 3.

x = 60°

That is,

the third angle = 60°

Then,

the first angle = 2x/3 = 2(60°) / 3 = 40°

the second angle = 4x/3 = 4(60°) / 3 = 80°

So, the three angles of the triangle are 40°, 80° and 60°.

**Problem 12 : **

In a triangle, sum of the first and second angles is 95°, sum of the second and third angles is 130° and sum of the first and third angles is 135°. Find the three angles of the triangle.

**Solution :**

Let x, y and z be the first, second and third angles of the triangle respectively.

From the given information, we have

x + y = 95° -----(1)

y + z = 130° -----(2)

x + z = 135° -----(3)

Add (1), (2) and (3) :

2x + 2y + 2z = 360°

2(x + y + z) = 360°

Divide each side by 2.

x + y + z = 180°

Substitute 95° for 'x + y' [from (1)].

95° + z = 180°

z = 85°

Substitute z = 85° in (2).

y + 85° = 130°

y = 45°

Substitute y = 45° in (1).

x + 45° = 95°

x = 50°

So, the three angles of the triangle are 50°, 45° and 85°.

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