If ABCDEF is a regular hexagon, then ` Delta `ACE is

Options

A. a right-angled triangle
B. an obtuse-angled triangle
C. an isosceles triangle
D. an equilateral triangle

Answer

To determine the nature of triangle ACE in a regular hexagon ABCDEF, we can follow these steps:

### Step 1: Understand the properties of a regular hexagon
A regular hexagon has six equal sides and six equal angles. The sum of the interior angles of a hexagon can be calculated using the formula:
\[ \text{Sum of interior angles} = (n - 2) \times 180^\circ \]
where \( n \) is the number of sides.

### Step 2: Calculate the sum of interior angles
For a hexagon, \( n = 6 \):
\[ \text{Sum of interior angles} = (6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ \]

### Step 3: Find the measure of each interior angle
Since all angles in a regular hexagon are equal:
\[ \text{Each interior angle} = \frac{720^\circ}{6} = 120^\circ \]

### Step 4: Analyze triangle ACE
In triangle ACE, we need to find the angles. We know:
- Angle ACB (which is the angle at vertex C) is one of the interior angles of the hexagon, so:
\[ \angle ACB = 120^\circ \]

### Step 5: Use the properties of triangle angles
In triangle ACE, let’s denote:
- Angle BAC = \( \theta \)
- Angle ACB = \( 120^\circ \)
- Angle EAC = \( \theta \)

Using the triangle angle sum property:
\[ \theta + \theta + 120^\circ = 180^\circ \]
This simplifies to:
\[ 2\theta + 120^\circ = 180^\circ \]

### Step 6: Solve for \( \theta \)
Subtract \( 120^\circ \) from both sides:
\[ 2\theta = 180^\circ - 120^\circ = 60^\circ \]
Dividing by 2 gives:
\[ \theta = 30^\circ \]

### Step 7: Determine the angles in triangle ACE
Thus, we have:
- Angle BAC = \( 30^\circ \)
- Angle ACB = \( 120^\circ \)
- Angle EAC = \( 30^\circ \)

### Step 8: Conclude the nature of triangle ACE
Since two angles (BAC and EAC) are equal (both \( 30^\circ \)), triangle ACE is an isosceles triangle. However, we also need to check the lengths of the sides. In a regular hexagon, the sides are equal, and since triangle ACE has angles of \( 30^\circ, 30^\circ, \) and \( 120^\circ \), it is not equilateral but isosceles.

### Final Conclusion
Triangle ACE is an isosceles triangle.

---