In an Isosceles triangle the base angles are equal and the vertex angle is twice of each base angle. Find the measures of the angles of the triangle.

Answer

To solve the problem, we need to find the measures of the angles in an isosceles triangle where the base angles are equal and the vertex angle is twice each base angle.

### Step-by-Step Solution:

1. **Define the Angles**:
Let the measure of each base angle (angle B and angle C) be \( x \). Since the vertex angle (angle A) is twice each base angle, we can express angle A as \( 2x \).

2. **Use the Triangle Sum Property**:
According to the triangle sum property, the sum of the angles in a triangle is always 180 degrees. Therefore, we can write the equation:
\[ \text{Angle A} + \text{Angle B} + \text{Angle C} = 180^\circ \]
Substituting the values we defined:
\[ 2x + x + x = 180^\circ \]

3. **Combine Like Terms**:
Simplifying the left side of the equation gives:
\[ 4x = 180^\circ \]

4. **Solve for \( x \)**:
To find the value of \( x \), divide both sides of the equation by 4:
\[ x = \frac{180^\circ}{4} = 45^\circ \]

5. **Find the Angles**:
Now that we have \( x \), we can find the measures of the angles:
- Angle B = \( x = 45^\circ \)
- Angle C = \( x = 45^\circ \)
- Angle A = \( 2x = 2 \times 45^\circ = 90^\circ \)

6. **Conclusion**:
The measures of the angles in the triangle are:
- Angle A (vertex angle) = \( 90^\circ \)
- Angle B (base angle) = \( 45^\circ \)
- Angle C (base angle) = \( 45^\circ \)

### Final Answer:
- Angle A = \( 90^\circ \)
- Angle B = \( 45^\circ \)
- Angle C = \( 45^\circ \)