Knowing one pair of sides of a quadrilateral to be congruent is not alone sufficient to prove that figure to be a parallelogram, nor is knowing one pair of sides of a quadrilateral to be parallel. But knowing both about the same pair of sides is sufficient by a theorem of parallelograms.

Consider the area formula for a trapezoid:

$\displaystyle A = \frac{1}{2} (B + b)h$

$\displaystyle \overline{MN}$ is the midsegment of the trapezoid, which, by the Trapezoid Midsegment Theorem, has length equal to the mean of the bases - in other words, $\displaystyle MN = \frac{1}{2} (B + b)$. The area formula can be expressed, after substitution, as 

$\displaystyle A = MN \cdot h$

So, if you know both the area and $\displaystyle MN$ - but not just one - you can find the height by dividing.

We can work backwards from the perimeter to find the length of the unknown side.

I) Gives us the perimeter.

II) Gives us two of the sides.

In a parallelogram there are two sets of corresponding sides. We can use I) and II) to write the following equaiton, where l is the length of our wanted side.

$\displaystyle 57=2\cdot13+2\cdot l$

Solve for l to find our final side:

$\displaystyle l=\frac{57-26}{2}=15.5$

Statement 1:  A circle with an area of $\displaystyle \pi$ is enclosed inside the square and touches all four edges of the square.

Write the formula for the area of the circle.

 $\displaystyle A=\pi r^2$

After substituting the value of the area and solve for radius, the radius is 1.  The side length of the square is double the length of the radius.

Statement 2:  A circle with a circumference of $\displaystyle \pi$ encloses the square and touches all four corners of the square.

Write the formula for the circumference of the circle.

$\displaystyle C=\pi D$

After substituting the value of the circumference and solve for diameter, the diameter of the circle is 1.  This diameter represents the diagonal of the square.  Because the square length and width are equal dimensions, it is possible to solve for the lengths of the square by Pythagorean Theorem knowing just the length of the square diagonal.

Therefore:

$\displaystyle \textup{EACH statement ALONE is sufficient.}$

Consider isosceles trapezoid $\displaystyle MNOP$.

I) $\displaystyle MNOP$ has a perimeter of $\displaystyle 360 megaparsecs$.

II) The larger base of $\displaystyle MNOP$ is 45 times bigger than the smaller base.

Find the length of the two legs of $\displaystyle MNOP$.

To find the length of the legs of a trapezoid, we need the perimeter and the two bases. Then, we can subtract the sum of the two bases from the perimeter and divide by two to find the lengths of the sides we are looking for.

Statement I gives us the perimeter of $\displaystyle MNOP$.

Statement II relates the two bases of $\displaystyle MNOP$.

We are told that $\displaystyle MNOP$ is an isosceles trapezoid, which means its two legs are equal; however, we still have too many unknowns and not enough equations. There is no way to solve this without more information.

$\displaystyle b_1+b_2+2l=360$

$\displaystyle b_2=45b_1$

We still have three unknowns and two equations, so we cannot solve this system of equations.

Each diagonal divides Rhombus $\displaystyle ABCD$ into two triangles, both isosceles.

Statement 1 alone establishes the perimeter of one such triangle. However, it does not make it clear what equal side lengths $\displaystyle AB$ and $\displaystyle AD$ and diagonal length $\displaystyle BD$ are. For example, $\displaystyle AB = AD = BD = 5$ fits the perimeter, but so does $\displaystyle AB = AD = 4, BD = 7$.

Statement 2 alone gives no information about the actual lengths of the sides.

Assume both statements are true. Since $\displaystyle \Delta ABC$ is equilateral, $\displaystyle m\angle ABC= 60 ^{\circ }$. It follows that $\displaystyle m\angle CDA = 60 ^{\circ }$, and $\displaystyle m \angle BCD = m \angle DAB = 120^{\circ }$. Also, the diagonals of a rhombus bisect their angles and are each other's perpendicular bisectors, so the rhombus, with their diagonals, is given below.

$\displaystyle \bigtriangleup ABD$ has perimeter $\displaystyle 15$, which means that 

$\displaystyle AB + BD + AD = 15$

$\displaystyle AB + BM + MD + AD = 15$

$\displaystyle AB + BM + BM + AB= 15$

$\displaystyle AB + BM = 7.5$

Since $\displaystyle \bigtriangleup ABD$ is known to be a $\displaystyle 30-60-90$ triangle, the proportions of the side lengths are known; along with the above equation, $\displaystyle AB$, and, subsequently, the perimeter, can be determined.

To find the perimeter of the quadrilateral, we need to know whether it is of a special type of quadrilaterals and we need to know the length of the sides.

Statement 1 tells us only that the quadrilateral is a rhombus. Indeed, a quadrilateral with perpendicular diagonals intersecting at their midpoint must be a rhombus. However we don't know any length of the sides. 

Statement 2 says gives us the length us two consecutive sides. It could be tempting to answer that it is sufficient, however, we can't conclude that the quadrilateral has equal lengths. Therefore this statement alone is insufficient.

Both statements together are sufficient since we can conclude that the quadrilateral is a rhombus, and twice  $\displaystyle (DB+BC)$ will give us the perimeter.

To find perimeter, we need to find the length of all the sides. Recall that rectangles are made up of two pairs of equal sides.

I) Relates one side to another non-equivalent side.

II) Gives us side $\displaystyle NT$, which must be equivalent to $\displaystyle CO$.

Use II) and I) to find all the side lengths, then add them up. Both are needed.

Recap:

Consider rectangle CONT

I) Side CO is three fourths of side ON

II) Side NT is 15.7 meters long

What is the perimeter of CONT? 

Because we are dealing with a rectangle, we know the following:

$\displaystyle CO=NT$              $\displaystyle ON=CT$

Find perimeter with:

$\displaystyle p=2l+2w$

Use I) and II) to write the following equation:

$\displaystyle 15.7=\frac{3}{4}ON$

So:

$\displaystyle ON=20.9\bar{3}$

And finally:

$\displaystyle p=2\cdot15.7+2\cdot20.9\bar{3}=73.2\bar{6}$

Statement 1): The area of the rectangle is 24.

Write the area for a rectangle and substitute the value of the area.

$\displaystyle A=L\cdot W$

$\displaystyle 24=L\cdot W$

The length and width of the rectangle are unknown, and each set of dimensions will provide a different perimeter.  This statement is insufficient to find the perimeter of the rectangle.

Statement 2): The diagonal of the rectangle is 5.

Given the diagonal of the rectangle, the Pythagorean Theorem can be used to solve for the diagonal.  Express the equation in terms of length and width.

$\displaystyle 5^2=L^2+W^2$

Similar to the case in Statement 1), both the length and width are unknown, and the equation by itself is insufficient to solve for the perimeter of the rectangle.

Attempting to use both equations: $\displaystyle 24=L\cdot W$ and $\displaystyle 5^2=L^2+W^2$ to solve for length and width will yield complex numbers as part of the solution.

Therefore:

$\displaystyle \textup{BOTH statements TOGETHER are NOT sufficient, and additional data is needed to answer the question. }$

Any two consecutive angles of a parallelogram are supplementary, so if one angle has measure $\displaystyle 90^{\circ }$, all angles can be proven to have measure $\displaystyle 90^{\circ }$. This is the definition of a rectangle. Statement 1 therefore proves the parallelogram to be a rectangle.

Also, a parallelogram is a rectangle if and only if its diagonals are congruent, which is what Statement 2 asserts. 

From either statement, it follows that parallelogram $\displaystyle ABCD$ is a rectangle.