ABCD is an isosceles trapezium. Prove that ΔADB ≅ ΔBCA. Hence deduce that the diagonals of an isosceles trapezium are equal

To prove that \( \Delta ADB \cong \Delta BCA \) for an isosceles trapezium ABCD, follow these steps:

### Given:
- ABCD is an **isosceles trapezium**, meaning that \( AB \parallel CD \) and \( AD = BC \).
- We are required to prove \( \Delta ADB \cong \Delta BCA \) and deduce that the diagonals \( AC \) and \( BD \) are equal.

### Step 1: Prove \( \Delta ADB \cong \Delta BCA \)

To prove that two triangles are congruent, we can use several criteria like **SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle),** or **AAS (Angle-Angle-Side)**. Here, we will use **SAS (Side-Angle-Side)**.

1. **Side 1 (AD = BC)**:
   - Since ABCD is an isosceles trapezium, the non-parallel sides are equal.
   - Therefore, \( AD = BC \).

2. **Angle 1 (\( \angle ADB = \angle BCA \))**:
   - \( AB \parallel CD \), and \( BD \) and \( AC \) are diagonals. 
   - \( \angle ADB \) and \( \angle BCA \) are **alternate interior angles** formed by the transversal \( BD \) and \( AC \).
   - Hence, \( \angle ADB = \angle BCA \).

3. **Side 2 (AB = AB)**:
   - \( AB \) is common to both \( \Delta ADB \) and \( \Delta BCA \).

Therefore, by **SAS** (Side-Angle-Side) congruence rule:
\[
\Delta ADB \cong \Delta BCA
\]

### Step 2: Deduce that the diagonals are equal

Since \( \Delta ADB \cong \Delta BCA \), the corresponding sides of the congruent triangles are equal. Therefore:
\[
AC = BD
\]
Thus, the diagonals of an isosceles trapezium are equal.

### Conclusion:
We have proved that \( \Delta ADB \cong \Delta BCA \), and as a result, the diagonals \( AC \) and \( BD \) of an isosceles trapezium are equal.

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Do you want more details on any part of this proof?

Here are 5 related questions for further exploration:

1. What other criteria can be used to prove triangles congruent besides SAS?
2. How would the proof change if the trapezium was not isosceles?
3. Can you use this congruence method to prove the equality of diagonals in other quadrilaterals?
4. What are some other properties of isosceles trapeziums besides equal diagonals?
5. How does the parallel nature of sides \( AB \) and \( CD \) affect the angles within the trapezium?

**Tip**: Always look for parallel lines in geometry problems, as they often lead to the use of corresponding or alternate angles in proofs.

ABCD is an isosceles trapezium. Prove that ΔADB ≅ ΔBCA. Hence deduce that the diagonals of an isosceles trapezium are equal.

Explore a detailed geometric proof showing how to prove that ΔADB ≅ ΔBCA in an isosceles trapezium ABCD using SAS congruence, and deduce that the diagonals AC and BD are equal. Suitable for students in Grades 9-11, covering triangle congruence and properties of trapeziums.

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