By definition a parallelogram has two sets of opposite sides that are congruent/parallel. However, to find the area of a paralleogram, you need to know the base and height lengths. Since this problem provides both the base and height measurements, apply the formula: 


Before applying the formula you must find  of . 

The solution is: 



Note: when working with multiples of ten remove zeros and then tack back onto the product.



There were two total zeros in the factors, so tack on two zeros to the product: 

The area of a triangle is 

Consider the rectangle ABCD

As a rectangle:

With  appearing directly in the center of the rectangle.

The area of 

Notice that the  term corresponds to the triangle's height.

The area of 

The two quantities are equal.

There are two components to solving this geometry puzzle. First, one must be aware that the sum of the measures of the interior angles of a parallelogram is 360 degrees (sum of the interior angles of a figure = 180(n-2), where n is the number of sides of the figure). Second, one must know that the other two interior angles are doubles of those given here. Thus if we assign one interior angle as x and the other as x-25, we find that x + x + (x-25) + (x-25) = 360. Combining like terms leads to the equation 4x-50=360. Solving for x we find that x = 410/4, 102.5, or approximately 103 degrees. Since x is the measure of the larger angle, this is our answer.

By using the properties of parallelograms along with those of supplementary angles, we can rewrite our figure as follows:

Recall, for example, that angle  is equal to:

, hence 

Now, you know that these angles can all be added up to . You should also know that 

Therefore, you can write:

Simplifying, you get:

Now, this means that:

 and . Thus, the two values are equal.

Because of the character of parallelograms, we know that our figure can be redrawn as follows:

Because it is a four-sided figure, we know that the sum of the angles must be . Thus, we know:

Solving for , we get:

In this problem, you are told that the sum of the two bases in a parallelogram is  Since the two bases must be equivalent, each side must equal: . Additionally, the problem states that the adjacent sides are  the length of the bases. Therefore, an adjacent side to the base must equal: 

To find one of the adjacent sides to the base, first note that the interior triangles represented by the red vertical lines must have a height of  and a base length of  Then, apply the formula:  to find the length of one side. 

Thus, the solution is: 









Therefore, the sum of the two sides is: 

A parallelogram must have two sets of congruent/parallel opposite sides. This parallelogram must have two sides with a measurement of  and two base sides each with a length of . In this question, you are provided with the information that the parallelogram has a base of  and a total perimeter of . Thus, work backwards using the perimeter formula in order to find the sum of the two adjacent sides to the base.

, where  and  are the measurements of adjacent sides. 

Thus, the solution is:

A parallelogram must have two sets of congruent/parallel opposite sides. Therefore, this parallelogram must have two sides with a measurement of  and two base sides each with a length of  Since the perimeter and one base length is provided in the question, work backwards using the perimeter formula:

, where  and  are the measurements of adjacent sides. 

Thus, the solution is:









Check: