Given a trapezoid ABCD with AD parallel to BC. The diagonals AC and BD are congruent. Prove that ABCD is an isosceles trapezoid.

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To prove that trapezoid ABCD is an isosceles trapezoid, we will use the fact that the diagonals AC and BD are congruent.

Let's start by drawing the trapezoid ABCD:

```
A ________ B
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D ________ C
```

Since AD is parallel to BC, we can conclude that angles DAB and ABC are congruent (corresponding angles). Similarly, angles BCD and CDA are congruent.

Now, let's consider triangles ABC and ADC. We know that the diagonals AC and BD are congruent, so AC ≅ BD.

In triangle ABC, we have:

- Side AB ≅ CD (a property of trapezoids)
- Angle ABC ≅ Angle CDA (corresponding angles)
- AC ≅ AC (common side between the two triangles)

According to the Side-Angle-Side (SAS) congruence theorem, triangles ABC and ADC are congruent.

Therefore, corresponding sides and angles in congruent triangles are congruent. By this theorem, we have:

AB ≅ DC (corresponding sides)
Angle BAC ≅ Angle DCA (corresponding angles)

Since AB ≅ DC and angle BAC ≅ DCA, trapezoid ABCD is an isosceles trapezoid.

Thus, we have proved that if the diagonals AC and BD of trapezoid ABCD are congruent, then ABCD is an isosceles trapezoid.