One exterior angle of a triangle is `70^(@)`. The two opposite interior angles are in the ratio 2:5. find the angles of the triangle.

Answer

To find the angles of the triangle given that one exterior angle is \(70^\circ\) and the two opposite interior angles are in the ratio \(2:5\), we can follow these steps:

### Step 1: Understand the relationship between the exterior angle and the interior angles
The exterior angle of a triangle is equal to the sum of the two opposite interior angles. In this case, we have:

\[ \text{Exterior angle} = 70^\circ \]

Let the two opposite interior angles be \(A\) and \(B\). According to the exterior angle property:

\[ A + B = 70^\circ \]

### Step 2: Express the interior angles in terms of a variable
Given that the two angles are in the ratio \(2:5\), we can express them as:

\[ A = 2x \quad \text{and} \quad B = 5x \]

### Step 3: Substitute the expressions into the equation
Now we can substitute \(A\) and \(B\) into the equation from Step 1:

\[ 2x + 5x = 70^\circ \]

### Step 4: Combine like terms
Combine the terms on the left side:

\[ 7x = 70^\circ \]

### Step 5: Solve for \(x\)
Now, divide both sides by \(7\):

\[ x = \frac{70^\circ}{7} = 10^\circ \]

### Step 6: Find the values of angles \(A\) and \(B\)
Now that we have \(x\), we can find the angles \(A\) and \(B\):

\[ A = 2x = 2 \times 10^\circ = 20^\circ \]
\[ B = 5x = 5 \times 10^\circ = 50^\circ \]

### Step 7: Find the third angle of the triangle
The third angle \(C\) can be calculated using the fact that the sum of the angles in a triangle is \(180^\circ\):

\[ C = 180^\circ - (A + B) = 180^\circ - (20^\circ + 50^\circ) = 180^\circ - 70^\circ = 110^\circ \]

### Final Angles of the Triangle
Thus, the angles of the triangle are:

\[ A = 20^\circ, \quad B = 50^\circ, \quad C = 110^\circ \]

### Summary of Angles
The angles of the triangle are \(20^\circ\), \(50^\circ\), and \(110^\circ\).

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