A Quadrilateral Is a Parallelogram
But first, let's go over five ways you can use to prove that a quadrilateral is a parallelogram. Depending on what information you have to work with, you'll be using one of these five ways.
1. Prove that both pairs of opposite sides are parallel.
This one is simply the reverse of the definition of a parallelogram. If you can prove that the quadrilateral fits the definition of a parallelogram, then it is a parallelogram.
2. Prove that both pairs of opposite sides are congruent.
If both pairs of opposite sides of a quadrilateral are congruent, then you'll always have two opposite pairs of parallel sides. Congruent means that they measure the same. Think about it: two congruent sides separating the other pair of opposite sides must always keep those opposite lines the same distance apart. This means, then, that the opposite sides are also parallel.
3. Prove that one pair of opposite sides is both congruent and parallel.
This one is kind of similar to the method before. It just goes about proving the case in another way. You can actually try this out with four toothpicks. Try it by placing two of the toothpicks opposite and parallel to each other. Now connect these two toothpicks at both ends with the other two toothpicks. You'll notice that no matter how you place your first pair of toothpicks, your second pair of toothpicks will always be parallel.
4. Prove that the diagonals bisect each other (that they can divide each other into two equal parts where they cross).
This one is a bit harder to visualize. But you can play around with it by taking two differentsized sticks and crossing them in the middle of both sticks. These two sticks are the diagonals inside your parallelogram. You'll see that no matter how you cross your sticks, as long as they cross in the middle, you'll always get a parallelogram. Your two sticks are the blue and green lines. You can see that these are the diagonals inside the parallelogram.

5. Prove that both pairs of opposite angles are congruent.
If both pairs of opposite angles are congruent, then your opposite pairs of sides will always be the same distance apart, thus making sure that they remain parallel and congruent. You can try this out by making two identical angles and then placing the two angles opposite each other so that the other pair of opposite angles are also congruent. Then you'll see that you'll always get a parallelogram.
Example
Now let's look at an example:
Prove that quadrilateral PORK is a parallelogram if triangle PRK is isosceles with base KR and triangle POR is also isosceles with base OP. Angle PRK is also congruent to angle RPO.
We begin by making our necessary marks to show our given information. Place tick marks on sides KP, PR, and OR to show that they are all the same, since both triangles are isosceles and they share a common side. An isosceles triangle is a triangle with two equal sides and a third side called the base. The two angles next to the base are also congruent. We also place congruent marks on angles K, PRK, O, and OPR.
Now we can go ahead with our proof.
Statement 
Reason 
Triangle PRK is isosceles with base KR 
Given 
Triangle ROP is isosceles with base OP 
Given 
Side PK = Side PR 
Triangle PRK is isosceles, therefore sides are congruent 
Side PR = Side PR 
Reflexive 
Side PR = Side OR 
Triangle POR is isosceles, therefore sides are congruent 
Angle PRK = Angle RPO 
Given 
Angle K = KRP 
Triangle PRK is isosceles, therefore base angles are congruent 
Angle O = Angle OPR 
Triangle POR is isosceles, therefore base angles are congruent 
Angle K = Angle O 
Transitive 
Triangle PRK = Triangle ROP 
AAS (Angle Angle Side Theorem) 
Angle KPR = Angle ORP 
Congruent angle of congruent triangles 
Angle KRO = Angle KPO 
Angle KRP + Angle PRO = Angle KPR + Angle OPR because Angle KRP = Angle OPR and Angle KPR = Angle PRO 
Quadrilateral PORK is a parallelogram 
Both pairs of opposite angles are congruent 
With this proof, we prove that the quadrilateral is a parallelogram by proving that both pairs of opposite angles are congruent.
Making your proof can take a while, and there is definitely more than one way to go about writing this proof. The most important thing for you to remember is that your proof needs to prove one of the five ways mentioned.
Lesson Summary
Let's review. A quadrilateral is a foursided flat shape. A parallelogram is a quadrilateral with two pairs of opposite and parallel sides.
To prove a quadrilateral is a parallelogram, you must use one of these five ways.
 Prove that both pairs of opposite sides are parallel.
 Prove that both pairs of opposite sides are congruent.
 Prove that one pair of opposite sides is both congruent and parallel.
 Prove that the diagonals bisect each other.
 Prove that both pairs of opposite angles are congruent.