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The diagram you have provided shows a triangle \( \triangle ABC \) with two perpendicular lines drawn from points \( E \) and \( F \), and segments marked, indicating symmetry. Here's the strategy to prove that \( \angle A \) is congruent to \( \angle C \).

### Steps for Proof:

1. **Isosceles Triangle Setup**:
   - The figure shows that segments \( AD \) and \( DC \) are marked as congruent (denoted by the equal markings). This means that \( D \) is the midpoint of \( AC \), so \( AD = DC \).
   - Additionally, \( BD \) is perpendicular to \( AC \), implying that \( \triangle ABD \) and \( \triangle CBD \) are mirror images of each other about \( BD \).

2. **Right Triangles**:
   - Triangles \( \triangle AEB \) and \( \triangle CFB \) contain right angles at \( E \) and \( F \), respectively, meaning these are right triangles. They share the side \( AB = BC \) (since it's also an isosceles triangle as marked), and the altitude \( BD \) bisects \( AC \), which further supports the symmetry between these triangles.

3. **Angle Correspondence**:
   - The fact that \( \triangle ABD \cong \triangle CBD \) (by Side-Angle-Side or SAS, since \( AD = DC \), \( BD \) is common, and the included angles \( \angle ABD \) and \( \angle CBD \) are both 90°) implies that:
     - \( \angle A = \angle C \).

Thus, we have shown that by symmetry and congruence of the right triangles in the diagram, \( \angle A \) is congruent to \( \angle C \).

Do you need further details on any specific part of the proof or explanation?

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### Related Questions:

1. What theorem justifies the congruence of two triangles when two sides and the included angle are equal?
2. Why is \( D \) the midpoint of \( AC \) in this diagram?
3. How does perpendicularity help prove congruence in this type of geometric problem?
4. What role does the isosceles property of \( \triangle ABC \) play in the proof?
5. How would the proof change if the triangle were not isosceles?

### Tip:
When proving congruence of angles or triangles, look for symmetry, midpoints, and perpendicular bisectors, as they often help identify corresponding congruent parts.

In this geometric proof, learn how to prove that angle A is congruent to angle C using the isosceles triangle theorem and symmetry in the triangle setup. This problem involves congruence and perpendicularity, suitable for students in Grades 9-11.

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