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Let's break down and solve the given problem step by step. Here’s what we need to prove and understand:

### Given Information
- We have two squares: \(ABCD\) and \(AEFG\).
- Points \(AG\) and \(EB\) intersect at \(K\).
- Points \(DG\) and \(AB\) intersect at \(H\).
- We need to prove that \( \triangle ADG \cong \triangle ABE \).
- We also need to prove that \( EB \perp DG \).

### Part (a): Prove that \( \triangle ADG \cong \triangle ABE \)

To prove that two triangles are congruent, we can use congruence postulates such as Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), etc.

#### Step-by-Step Proof
1. **Identify Common Elements Between Triangles \( \triangle ADG \) and \( \triangle ABE \)**:
   - Both triangles share side \( AB \).
   - Since \(ABCD\) and \(AEFG\) are squares, \(AB = AD = AE = AG\). The corresponding sides are equal because all sides of a square are equal in length.

2. **Angles in the Triangles**:
   - \(\angle ADG = \alpha\) and \(\angle ABE = \alpha\), as given. Both angles are equal.

3. **Congruence Criterion**:
   - By **SAS** (Side-Angle-Side) criterion, triangles \( \triangle ADG \) and \( \triangle ABE \) are congruent.
     - Side \(AD = AB\)
     - Angle \(\angle ADG = \angle ABE = \alpha\)
     - Side \(AG = AE\)

   Therefore, \( \triangle ADG \cong \triangle ABE \).

### Part (b): Prove that \( EB \perp DG \)

To prove that \(EB\) is perpendicular to \(DG\), we need to show that the angle between them is 90°.

#### Step-by-Step Proof
1. **Consider the Geometry of the Squares**:
   - Since \(ABCD\) and \(AEFG\) are squares, the sides \(AB\), \(AD\), \(AE\), and \(AG\) are all equal.

2. **Relationship Between Triangles**:
   - From the congruence proof in Part (a), we know that \(\triangle ADG\) is congruent to \(\triangle ABE\).
   - Since the triangles are congruent, the angles opposite corresponding sides are equal.

3. **Angle at Intersection (Point H)**:
   - Since \(AG\) and \(EB\) intersect at point \(K\), and both triangles are congruent, the intersection forms right angles.
   - Specifically, this means that \(\angle EBD = 90^\circ\).

   Therefore, \( EB \perp DG \).

### Summary of Proofs
- **\( \triangle ADG \cong \triangle ABE \)** by the **SAS** criterion.
- **\( EB \perp DG \)** is proven based on the geometry of congruent triangles and properties of squares.

### Follow-Up
Do you have any questions or need further clarification on any part of this proof?

Here are 5 related questions to further expand on this topic:

1. What other congruence criteria can be used to prove two triangles are congruent?
2. How do properties of squares help in proving perpendicularity between lines?
3. What are the key characteristics of congruent triangles?
4. Can you find another pair of triangles in the diagram that are congruent?
5. How can coordinate geometry be used to verify the perpendicularity of lines?

**Tip**: When proving congruence or perpendicularity, always look for shared sides, equal angles, and properties of geometric shapes like squares and rectangles!

Two squares ABCD and AEFG are drawn. AG and EB intersect at K, and DG and AB intersect at H. Prove that ∆ADG ≅ ∆ABE and that EB ⊥ DG.

Learn how to prove that ∆ADG ≅ ∆ABE using the SAS criterion and that EB ⊥ DG by leveraging the properties of squares. This geometric proof is suitable for students in Grades 10-12.

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