To prove that line AB is equal to line CD and line BC is equal to line AD in parallelogram ABCD, we can use the properties of parallelograms.
1. Opposite sides of a parallelogram are parallel.
Since ABCD is a parallelogram, we know that AB is parallel to CD and BC is parallel to AD.
2. If two parallel lines are intersected by a transversal, then the alternate interior angles are congruent.
Using the fact that AB is parallel to CD, we can draw a transversal line, say line EF, intersecting these parallel lines at points E and F. This gives us alternate interior angles AEF and CFE, which are congruent. Similarly, using the fact that BC is parallel to AD, we can draw another transversal line, say line GH, intersecting these parallel lines at points G and H. This gives us alternate interior angles CBG and DAG, which are congruent.
3. If two angles of a parallelogram are congruent, then the other two angles are also congruent.
From step 2, we know that angle AEF is congruent to angle CFE and angle CBG is congruent to angle DAG. Using the fact that the opposite angles of a parallelogram are congruent, we can say that angle ABD is congruent to angle CDA and angle BCD is congruent to angle DAB.
4. If two angles of a quadrilateral are congruent, then the opposite sides of the quadrilateral are congruent.
Using step 3, we know that angle ABD is congruent to angle CDA and angle BCD is congruent to angle DAB. Therefore, we can say that AB is congruent to CD and BC is congruent to AD.
Therefore, we have proved that line AB is equal to line CD and line BC is equal to line AD in parallelogram ABCD.