The diagonals of an isosceles trapezoid divided it into one pair of congruent triangles and one pair of similar triangles.

To prove that the diagonals of an isosceles trapezoid divide it into one pair of congruent triangles and one pair of similar triangles, we can use the properties of triangles and trapezoids.

Here's how to prove it:

1. Start by drawing an isosceles trapezoid. An isosceles trapezoid has two parallel sides, where the shorter side lengths are congruent to each other (let's call them AB and CD), and the longer side lengths are also congruent to each other (let's call them BC and AD). Also, draw the diagonal connecting the non-parallel sides (AC).

2. Label the points where the diagonals intersect the base as E and F, with E on side AD and F on side BC. This means that AC is the diagonal that divides the isosceles trapezoid into two smaller triangles, AEC and BFC.

3. To prove that AEC and BFC are congruent triangles, we need to show that their corresponding sides and angles are congruent.

4. First, note that AE = CF because they are both diagonals of the trapezoid.

5. Secondly, since AB || CD (parallel sides), we have ∠A = ∠C and ∠B = ∠D (corresponding angles).

6. Similarly, since AD || BC, we have ∠AED = ∠CFB and ∠DAE = ∠CBF (alternate interior angles).

7. Now we can apply the Side-Angle-Side (SAS) congruence postulate to show that AEC and BFC are congruent triangles:
a) Side: AE = CF (already proved in step 4)
b) Angle: ∠A = ∠C and ∠B = ∠D (already proved in step 5)
c) Side: AC (common side)

8. Therefore, we can conclude that AEC and BFC are congruent triangles.

9. Next, to prove that AEC and BFC are similar triangles, we need to show that their corresponding angles are congruent.

10. Since ∠A = ∠C and ∠B = ∠D (as proved in step 5), and both pairs of triangles share angle ∠AEC = ∠BFC (vertical angles), we can conclude that AEC and BFC are similar triangles.

In summary, we have proven that the diagonal AC of the isosceles trapezoid divides it into one pair of congruent triangles (AEC and BFC) and one pair of similar triangles (AEC and BFC).