Final answer:
In a parallelogram, opposite sides are equal in length, so in parallelogram ABCD, AB = CD and BC = DA. This can be proven using vector addition and subtraction, along with the properties of similar and congruent triangles pertinent to parallelograms.
Step-by-step explanation:
The student's question is focused on proving the properties of sides within a parallelogram.
In a parallelogram, opposite sides are equal in length.
Therefore, in parallelogram ABCD, AB = CD and BC = DA.
To prove these properties, one can use vector addition and subtraction as well as geometric congruence.
By constructing a parallelogram with vectors and measuring diagonals and angles as described, one can deduce the lengths of sides and relationships between them.
For example, using similar triangles and congruent triangles helps to establish these equalities between opposite sides.
As we align vectors and use congruent triangles and properties specific to parallelograms such as their equal opposite sides and angles, it becomes apparent that these sides are indeed equal as per the properties that define parallelograms.