State, true or false :
Two isosceles triangles are similar, if an angle of one is congruent to the corresponding angle of the other. 

Answer

To determine whether the statement "Two isosceles triangles are similar if an angle of one is congruent to the corresponding angle of the other" is true or false, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definition of Isosceles Triangles**:
- An isosceles triangle has at least two sides that are equal in length, which also means that the angles opposite those sides are equal.

2. **Draw Two Isosceles Triangles**:
- Let’s denote the first isosceles triangle as \( \triangle ABC \) and the second as \( \triangle DEF \).

3. **Identify the Congruent Angles**:
- According to the problem, we assume that angle \( A \) in triangle \( ABC \) is congruent to angle \( D \) in triangle \( DEF \). Thus, we can write:
\[ \angle A = \angle D \]

4. **Determine the Other Angles**:
- Since \( \triangle ABC \) is isosceles, the angles opposite the equal sides (let's say \( B \) and \( C \)) must be equal:
\[ \angle B = \angle C \]
- Similarly, for triangle \( DEF \), the angles opposite the equal sides (let's say \( E \) and \( F \)) must also be equal:
\[ \angle E = \angle F \]

5. **Assign Values to the Angles**:
- Let’s assume \( \angle A = \angle D = 40^\circ \).
- For triangle \( ABC \):
\[ \angle B + \angle C + \angle A = 180^\circ \implies \angle B + \angle C + 40^\circ = 180^\circ \]
\[ \angle B + \angle C = 140^\circ \]
Since \( \angle B = \angle C \):
\[ 2\angle B = 140^\circ \implies \angle B = 70^\circ \quad \text{and} \quad \angle C = 70^\circ \]

- For triangle \( DEF \):
\[ \angle E + \angle F + \angle D = 180^\circ \implies \angle E + \angle F + 40^\circ = 180^\circ \]
\[ \angle E + \angle F = 140^\circ \]
Since \( \angle E = \angle F \):
\[ 2\angle E = 140^\circ \implies \angle E = 70^\circ \quad \text{and} \quad \angle F = 70^\circ \]

6. **Establish Corresponding Angles**:
- Now we have:
\[ \angle A = \angle D = 40^\circ, \quad \angle B = \angle E = 70^\circ, \quad \angle C = \angle F = 70^\circ \]

7. **Apply the Angle-Angle (AA) Similarity Criterion**:
- Since we have two pairs of corresponding angles that are equal:
\[ \angle A = \angle D, \quad \angle B = \angle E, \quad \angle C = \angle F \]
- By the AA criterion, we can conclude that:
\[ \triangle ABC \sim \triangle DEF \]

8. **Conclusion**:
- Therefore, the statement is **True**: Two isosceles triangles are similar if an angle of one is congruent to the corresponding angle of the other.