One of the angle of a triangle is equal to the sum of the other two angles. If the ratio of the other two angles is 4:5, then the angles of triangle are

Options

A. `40^(@), 50^(@), 90^(@)`
B. `40^(@), 60^(@), 80^(@)`
C. `50^(@) , 50^(@), 80^(@)`
D. None of these

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the Angles
Let the two angles of the triangle be \( \theta_1 \) and \( \theta_2 \). According to the problem, the ratio of these angles is given as \( 4:5 \).

### Step 2: Express Angles in Terms of a Variable
Let \( \theta_1 = 4x \) and \( \theta_2 = 5x \), where \( x \) is a common variable.

### Step 3: Find the Third Angle
The third angle \( \theta_3 \) is equal to the sum of the other two angles:
\[ \theta_3 = \theta_1 + \theta_2 = 4x + 5x = 9x \]

### Step 4: Use the Triangle Angle Sum Property
The sum of the angles in a triangle is always \( 180^\circ \):
\[ \theta_1 + \theta_2 + \theta_3 = 180^\circ \]
Substituting the expressions for the angles:
\[ 4x + 5x + 9x = 180^\circ \]
This simplifies to:
\[ 18x = 180^\circ \]

### Step 5: Solve for \( x \)
Now, divide both sides by 18:
\[ x = \frac{180^\circ}{18} = 10^\circ \]

### Step 6: Calculate Each Angle
Now we can find the values of \( \theta_1 \), \( \theta_2 \), and \( \theta_3 \):
- \( \theta_1 = 4x = 4 \times 10^\circ = 40^\circ \)
- \( \theta_2 = 5x = 5 \times 10^\circ = 50^\circ \)
- \( \theta_3 = 9x = 9 \times 10^\circ = 90^\circ \)

### Final Answer
Thus, the angles of the triangle are:
- \( \theta_1 = 40^\circ \)
- \( \theta_2 = 50^\circ \)
- \( \theta_3 = 90^\circ \)

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