https://file.aiquickdraw.com/mathbot/file/1731672283310odit4jkb.jpg

The problem in the image involves an isosceles trapezium \(ABCD\) where \(AB \parallel DC\), and we need to prove that \(AC = BD\).

### Solution:
In an isosceles trapezium:
1. The non-parallel sides (legs) are equal in length.
2. The diagonals are also equal in length.

Here’s a step-by-step proof:

1. **Identify Known Properties**: Since \(AB \parallel DC\) and \(ABCD\) is isosceles, we have \(AD = BC\) by the definition of an isosceles trapezium.
   
2. **Parallel Line Implication**: In isosceles trapeziums, when the pair of opposite sides are parallel and the other pair of sides are equal, it results in the diagonals \(AC\) and \(BD\) being equal in length.

3. **Use Symmetry**: Due to the symmetry of the isosceles trapezium, the diagonals \(AC\) and \(BD\) mirror each other, ensuring that they are congruent.

Thus, we conclude:
\[
AC = BD
\]

Would you like further details on any specific step, or do you have other questions?

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Here are five related questions to expand your understanding:

1. How can we use triangle congruence criteria to further justify the equality of the diagonals in an isosceles trapezium?
2. What other properties do diagonals of an isosceles trapezium possess?
3. How does the parallelism in a trapezium help in solving geometric proofs?
4. How would the proof change if \(ABCD\) were not isosceles?
5. What is the relationship between the angles in an isosceles trapezium?

**Tip:** Remember that symmetry in geometric shapes often simplifies proofs, especially in isosceles shapes where both sides and angles have mirrored properties.

In an isosceles trapezium with AB || DC, prove that AC = BD.

Explore a geometric proof to show that the diagonals in an isosceles trapezium with parallel sides AB and DC are equal. This problem involves understanding the properties of isosceles trapeziums and the symmetry of their diagonals. Ideal for students in Grades 9-10.

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