An exterior angle is drawn to a triangle, which is acute , then on the basis of angles what type of triangle is this -

Options

A. Acute angled
B. Obtuse angled
C. Right angled
D. Scalene

Answer

To determine the type of triangle based on the given acute exterior angle, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Exterior Angle**:
- Let's denote the triangle as \( ABC \) and the exterior angle at vertex \( C \) as \( y \).
- By definition, the exterior angle \( y \) is equal to the sum of the two opposite interior angles, which are \( \angle ABC \) and \( \angle BAC \).
- Therefore, we can write:
\[ y = \angle ABC + \angle BAC \]

2. **Given Information**:
- We know that the exterior angle \( y \) is acute, meaning:
\[ 0^\circ < y < 90^\circ \]

3. **Using the Triangle Angle Sum Property**:
- The sum of the angles in triangle \( ABC \) is:
\[ \angle ABC + \angle BAC + \angle ACB = 180^\circ \]
- Rearranging this gives us:
\[ \angle ABC + \angle BAC = 180^\circ - \angle ACB \]

4. **Substituting the Exterior Angle**:
- Now, substituting the expression for \( \angle ABC + \angle BAC \) into the equation for \( y \):
\[ y = 180^\circ - \angle ACB \]

5. **Setting Up the Inequality**:
- Since \( y \) is acute, we have:
\[ 0^\circ < 180^\circ - \angle ACB < 90^\circ \]
- This leads to two inequalities:
- From \( 0^\circ < 180^\circ - \angle ACB \), we get:
\[ \angle ACB < 180^\circ \]
- From \( 180^\circ - \angle ACB < 90^\circ \), we can rearrange it to:
\[ \angle ACB > 90^\circ \]

6. **Conclusion**:
- Since \( \angle ACB \) is greater than \( 90^\circ \), it indicates that triangle \( ABC \) has one angle that is obtuse.
- Therefore, triangle \( ABC \) is classified as an **obtuse triangle**.

### Final Answer:
The triangle is an **obtuse triangle**.

---