The exterior angle of a triangle is `105^(@)` and one of the interior opposite angles is `60^(@)`. Find the other two angles.

Options

A. `45^(@),45^(@)`.
B. `25^(@),75^(@)`.
C. `45^(@),55^(@)`.
D. `45^(@),75^(@)`.

Answer

To solve the problem, we need to find the two unknown angles of the triangle given that one of the interior opposite angles is \(60^\circ\) and the exterior angle is \(105^\circ\).

### Step-by-Step Solution:

1. **Understanding the Exterior Angle**:
- We know that the exterior angle of a triangle is equal to the sum of the two opposite interior angles.
- Given the exterior angle \( \angle QPR = 105^\circ \) and one of the opposite interior angles \( \angle RPS = 60^\circ \).

**Hint**: Remember that the exterior angle is formed by extending one side of the triangle.

2. **Using the Exterior Angle Property**:
- According to the property of triangles, we can write:
\[ \angle QPR = \angle RPS + \angle QRP \]
- Substituting the known values:
\[ 105^\circ = 60^\circ + \angle QRP \]

**Hint**: This property states that the exterior angle is equal to the sum of the two opposite interior angles.

3. **Finding the Angle \( \angle QRP \)**:
- Rearranging the equation to find \( \angle QRP \):
\[ \angle QRP = 105^\circ - 60^\circ \]
\[ \angle QRP = 45^\circ \]

**Hint**: Subtract the known interior angle from the exterior angle to find the other interior angle.

4. **Finding the Third Angle \( \angle PQR \)**:
- Now we know two angles of the triangle: \( \angle RPS = 60^\circ \) and \( \angle QRP = 45^\circ \).
- We can use the angle sum property of triangles which states that the sum of all interior angles is \(180^\circ\):
\[ \angle PQR + \angle RPS + \angle QRP = 180^\circ \]
- Substituting the known values:
\[ \angle PQR + 60^\circ + 45^\circ = 180^\circ \]
- Simplifying this:
\[ \angle PQR + 105^\circ = 180^\circ \]
- Solving for \( \angle PQR \):
\[ \angle PQR = 180^\circ - 105^\circ \]
\[ \angle PQR = 75^\circ \]

**Hint**: Use the total sum of angles in a triangle to find the missing angle.

5. **Final Result**:
- The two angles we found are:
- \( \angle QRP = 45^\circ \)
- \( \angle PQR = 75^\circ \)

### Summary of Angles:
- The angles of the triangle are:
- \( \angle RPS = 60^\circ \)
- \( \angle QRP = 45^\circ \)
- \( \angle PQR = 75^\circ \)