Find the angles of a triangle whose interior and exterior angles are in the ratio of `2:7`. also, the other two interior angles are in the ratio of `3:4`.

Options

A. `40^(@),60^(@),80^(@)`.
B. `40^(@),40^(@),60^(@)`.
C. `40^(@),70^(@),70^(@)`.
D. `30^(@),70^(@),80^(@)`.

Answer

To find the angles of a triangle whose interior and exterior angles are in the ratio of 2:7, and where the other two interior angles are in the ratio of 3:4, we can follow these steps:

### Step 1: Define the angles based on the given ratios.
Let the interior angle be represented as \(2x\) and the exterior angle as \(7x\).

### Step 2: Use the property of interior and exterior angles.
According to the property of triangles, the sum of an interior angle and its corresponding exterior angle is \(180^\circ\). Therefore, we can write the equation:
\[ 2x + 7x = 180^\circ \]

### Step 3: Solve for \(x\).
Combine the terms:
\[ 9x = 180^\circ \]
Now, divide both sides by 9:
\[ x = \frac{180^\circ}{9} = 20^\circ \]

### Step 4: Find the measure of the interior angle.
Substituting \(x\) back into the expression for the interior angle:
\[ \text{Interior angle} = 2x = 2 \times 20^\circ = 40^\circ \]

### Step 5: Find the measure of the exterior angle.
Now, find the exterior angle:
\[ \text{Exterior angle} = 7x = 7 \times 20^\circ = 140^\circ \]

### Step 6: Set up the ratios for the other two interior angles.
Let the other two interior angles be \(3y\) and \(4y\) based on the given ratio of \(3:4\).

### Step 7: Use the angle sum property of triangles.
The sum of all interior angles in a triangle is \(180^\circ\):
\[ 40^\circ + 3y + 4y = 180^\circ \]
Combine the terms:
\[ 40^\circ + 7y = 180^\circ \]

### Step 8: Solve for \(y\).
Subtract \(40^\circ\) from both sides:
\[ 7y = 180^\circ - 40^\circ = 140^\circ \]
Now, divide both sides by 7:
\[ y = \frac{140^\circ}{7} = 20^\circ \]

### Step 9: Find the measures of angles B and C.
Now substitute \(y\) back to find angles B and C:
\[ \text{Angle B} = 3y = 3 \times 20^\circ = 60^\circ \]
\[ \text{Angle C} = 4y = 4 \times 20^\circ = 80^\circ \]

### Step 10: Summary of angles.
The angles of the triangle are:
- Angle A (interior angle) = \(40^\circ\)
- Angle B = \(60^\circ\)
- Angle C = \(80^\circ\)

### Final Check:
To verify, check if the sum of the angles equals \(180^\circ\):
\[ 40^\circ + 60^\circ + 80^\circ = 180^\circ \]
Thus, the solution is correct.