Given Parallelogram ABCD with Diagonal AC Drawn

In the given parallelogram ABCD, we are asked to prove that it is a parallelogram. A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.

To prove this, let's start by drawing a diagonal AC in the parallelogram ABCD as shown in the figure.

Since AB || DC and AD || BC, we can conclude that opposite angles are equal, i.e., ∠A = ∠C and ∠B = ∠D. This is because whenever alternate angles are equal, the lines are parallel.

Now, let's draw a line through E, the midpoint of AD, parallel to AB. This line intersects BC at F as shown in the figure.

Since PQ || BC and CR || BA, we can conclude that opposite sides are parallel, i.e., PR || BC and CR || BP.

Therefore, each pair of opposite sides is parallel, making ABCD a parallelogram.

Additionally, we know that diagonals of a parallelogram bisect each other. In this case, AC bisects ∠A as well as ∠C.

Furthermore, if the diagonals of a parallelogram are equal, then it is a rectangle. This can be proved by drawing two points P and Q on diagonal BD such that DP = BQ. Then, we can show that ABCD is a square or rhombus depending on the conditions.

In conclusion, given parallelogram ABCD with diagonal AC drawn, we have proved that:

* Opposite sides are parallel

* Diagonals bisect each other

* If diagonals are equal, it is a rectangle

* It can be a square or rhombus depending on the conditions

Thus, the given parallelogram ABCD with diagonal AC drawn satisfies all the properties of a parallelogram.

Given parallelogram abcd with diagonal ac drawn