ABCD is a quadrilateral in which AB parallel DC and AD=BC.prove that (1)angle A + angle C =180 degree = angle B + angle D (2)AC=BD

To prove both statements, we will use the properties of parallel lines and congruent triangles. Here's how you can prove them:

(1) Proving angle A + angle C = 180 degrees = angle B + angle D:
Since AB is parallel to DC, we can conclude that angle A + angle B = 180 degrees (as corresponding angles formed by a transversal intersecting parallel lines are supplementary).
Similarly, we can conclude that angle C + angle D = 180 degrees (as corresponding angles formed by a transversal intersecting parallel lines are supplementary).

Now we need to prove that angle A + angle C = angle B + angle D:

Given that AD = BC and AB is parallel to DC, we can conclude that triangles ADB and CBA are congruent by the Side-Angle-Side (SAS) criterion.

By congruence, we know that:
- angle A = angle C (corresponding angles of congruent triangles).
- angle D = angle B (corresponding angles of congruent triangles).

Therefore, angle A + angle C = angle C + angle C (using congruence), and angle B + angle D = angle B + angle B (using congruence).
Because angle B + angle B = angle C + angle C, we can substitute this in the equation to get:
angle A + angle C = angle B + angle D.

Hence, we have proved that angle A + angle C = 180 degrees = angle B + angle D.

(2) Proving AC = BD:
Since triangles ADB and CBA are congruent (as proven above), we know that:
- AD = BC (given)
- AB = CD (opposite sides of a parallelogram are congruent)

Now, by congruence, we can further deduce that AC = BD (using the Side-Side-Side (SSS) criterion for congruent triangles).
Therefore, we have proved that AC = BD.

Note: In both proofs, it is important to mention the given conditions and use relevant congruence postulates or theorems to justify the statements.