An exterior angle of a triangle is `115^@` and one of the interior opposite angles is `45^@`. Then the other two angles are

Options

A. `65^@, 70^@`
B. `60^@, 75^@`
C. `45^@, 90^@`
D. `50^@, 85^@`

Answer

To solve the problem, we need to find the two unknown angles of a triangle given that one of the interior opposite angles is \(45^\circ\) and the exterior angle is \(115^\circ\).

### Step-by-Step Solution:

1. **Understanding the Exterior Angle**:
The exterior angle of a triangle is equal to the sum of the two opposite interior angles. In this case, the exterior angle is \(115^\circ\) and one of the opposite interior angles is \(45^\circ\).

\[ \text{Exterior Angle} = \text{Interior Angle 1} + \text{Interior Angle 2} \]

Let’s denote the unknown interior angle as \(x\). Therefore, we can write the equation:

\[ 115^\circ = 45^\circ + x \]

2. **Solving for the Unknown Angle**:
To find \(x\), we can rearrange the equation:

\[ x = 115^\circ - 45^\circ \]

Now, performing the subtraction:

\[ x = 70^\circ \]

So, one of the interior angles is \(70^\circ\).

3. **Finding the Third Angle**:
We know that the sum of all angles in a triangle is \(180^\circ\). Thus, we can find the third angle \(y\) using the following equation:

\[ 45^\circ + 70^\circ + y = 180^\circ \]

Simplifying this gives:

\[ 115^\circ + y = 180^\circ \]

Now, isolating \(y\):

\[ y = 180^\circ - 115^\circ \]

Performing the subtraction:

\[ y = 65^\circ \]

4. **Final Angles**:
Therefore, the two interior angles of the triangle are \(70^\circ\) and \(65^\circ\).

### Summary of Angles:
- One interior angle: \(45^\circ\)
- Second interior angle: \(70^\circ\)
- Third interior angle: \(65^\circ\)

### Final Answer:
The other two angles are \(70^\circ\) and \(65^\circ\).

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