prove that two triangle formed by the diagonal of a parallelogram are congruent

use the SSS property, since the diagonal is common to both triangles, and the parallel sides are congruent.

To prove that the two triangles formed by the diagonal of a parallelogram are congruent, we can use the concept of parallel lines and the properties of a parallelogram.

Here's the step-by-step proof:

Step 1: Let's start by labeling the parallelogram. Draw a parallelogram ABCD, where AB is parallel to CD and AD is parallel to BC.

Step 2: Draw the diagonal AC, which divides the parallelogram into two triangles, say triangle ACD and triangle ABC.

Step 3: We need to show that triangle ACD is congruent to triangle ABC.

Step 4: Since AB is parallel to CD and AC is a transversal, we have alternate interior angles. This means that angle CAD is congruent to angle CBA (as they are corresponding angles).

Step 5: Similarly, since AD is parallel to BC and AC is a transversal, we have alternate interior angles. This means that angle ACD is congruent to angle ABC (as they are corresponding angles).

Step 6: Now, we know that angle CAD is congruent to angle CBA and angle ACD is congruent to angle ABC. Moreover, AC is common to both triangles.

Step 7: According to the Angle-Side-Angle (ASA) congruence condition, if two angles and the included side of one triangle are congruent to the corresponding two angles and the included side of another triangle, the two triangles are congruent.

Step 8: Applying the ASA congruence condition, we can conclude that triangle ACD is congruent to triangle ABC.

Hence, the two triangles formed by the diagonal of a parallelogram are proven to be congruent using the concept of parallel lines and the properties of a parallelogram.