Write a proof for the following

Given: line AC is congruent to line BD, line AD is congruent to BC
Prove: triangle DAX is congruent to triangle CBX

It's an isosceles trapezoid. From top left is labeled A. Top right labeled B. Bottom right labeled C. Bottom left labeled D. With X as the angle of AC and BD.

To prove that triangle DAX is congruent to triangle CBX, we can use the Side-Angle-Side (SAS) congruence criterion.

Here are the steps to prove the congruence:

1. Given information: We are given that line AC is congruent to line BD (AC ≅ BD) and line AD is congruent to BC (AD ≅ BC).

2. Draw the figure: Draw a trapezoid with vertices labeled as A, B, C, and D. The angle at the intersection of lines AC and BD is labeled as X.

3. Identify congruent sides: We know that AC ≅ BD and AD ≅ BC.

4. Identify shared angle: Both triangles DAX and CBX share the angle, X.

5. Apply SAS congruence criterion: According to the SAS criterion, two triangles are congruent if two sides and the included angle of one triangle are congruent to the corresponding sides and angle of the other triangle. In our case, we have:

Triangle DAX: DA ≅ CB (given), AX (common side), and ∠DAX = ∠CBX (both are labeled as X).

Triangle CBX: CB ≅ DA (given), BX (common side), and ∠CBX = ∠DAX (both are labeled as X).

So, by the SAS criterion, triangle DAX is congruent to triangle CBX.

Therefore, we have proved that triangle DAX is congruent to triangle CBX using the given information and the SAS congruence criterion.

To prove that triangle DAX is congruent to triangle CBX, we can use the SAS (Side-Angle-Side) congruence criterion.

Given:
1. Line AC is congruent to line BD.
2. Line AD is congruent to line BC.
3. Angles DAX and CBX are congruent (given that X is the angle of AC and BD).

Proof:
1. Since line AC is congruent to line BD (given), we have AC ≅ BD (by the definition of congruent segments).
2. Similarly, since line AD is congruent to line BC (given), we have AD ≅ BC.
3. By the transitive property of congruence, we can equate the corresponding parts: AC ≅ BD ≅ AD ≅ BC.
4. Angle DAX ≅ Angle CBX (given).
5. By the SAS congruence criterion, triangle DAX is congruent to triangle CBX, since we have two pairs of corresponding sides and the included angle congruent (AC ≅ BD, AD ≅ BC, Angle DAX ≅ Angle CBX).

Therefore, triangle DAX is congruent to triangle CBX.