In this problem, you are told that the sum of the two bases in a parallelogram is \(\displaystyle 30.\) Since the two bases must be equivalent, each side must equal: \(\displaystyle \frac{30}{2}=15\). Additionally, the problem states that the adjacent sides are \(\displaystyle \frac{1}{3}\) the length of the bases. Therefore, an adjacent side to the base must equal: 

\(\displaystyle 15\times \frac{1}{3}=\frac{15}{3}=5\)

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 \(\displaystyle 5\) and a base length of \(\displaystyle 10-8=2.\) Then, apply the formula: \(\displaystyle a^2+b^2=c^2\) to find the length of one side. 

Thus, the solution is: 

\(\displaystyle 2^2+5^2=c^2\)

\(\displaystyle 4+25=c^2\)

\(\displaystyle 29=c^2\)

\(\displaystyle c=\sqrt{29}\)

Therefore, the sum of the two sides is: 

\(\displaystyle \sqrt{29}+\sqrt{29}=2\sqrt{29}\)

A parallelogram must have two sets of congruent/parallel opposite sides. This parallelogram must have two sides with a measurement of \(\displaystyle 65\) and two base sides each with a length of \(\displaystyle 74\). In this question, you are provided with the information that the parallelogram has a base of \(\displaystyle 74\) and a total perimeter of \(\displaystyle 278\). Thus, work backwards using the perimeter formula in order to find the sum of the two adjacent sides to the base.

\(\displaystyle \textup{Perimeter}=2(a+b)\), where \(\displaystyle a\) and \(\displaystyle b\) are the measurements of adjacent sides. 

Thus, the solution is:

\(\displaystyle 278=2(74+b)\)

\(\displaystyle 278=148+2b\)

\(\displaystyle 2b=278-148=130\)

A parallelogram must have two sets of congruent/parallel opposite sides. Therefore, this parallelogram must have two sides with a measurement of \(\displaystyle 8\) and two base sides each with a length of \(\displaystyle 7.\) Since the perimeter and one base length is provided in the question, work backwards using the perimeter formula:

\(\displaystyle \textup{Perimeter}=2(a+b)\), where \(\displaystyle a\) and \(\displaystyle b\) are the measurements of adjacent sides. 

Thus, the solution is:

\(\displaystyle 30=2(7+b)\)

\(\displaystyle 30=14+2b\)

\(\displaystyle 2b=30-14=16\)

\(\displaystyle b=\frac{16}{2}=8\)

Check:

\(\displaystyle 30=7+7+8+8\)

\(\displaystyle 30=14+16\)

\(\displaystyle 30=30\)

A parallelogram must have two sets of congruent/parallel opposite sides. This parallelogram must have two sides with a measurement of \(\displaystyle 16\textup{mm}\) and two base sides each with a length of \(\displaystyle 24\textup{mm}\). In this question, you are provided with the information that the parallelogram has a base of \(\displaystyle 24\textup{mm}\) and a total perimeter of \(\displaystyle 80\textup{mm}\). Thus, work backwards using the perimeter formula in order to find the length of one missing side that is adjacent to the base.

\(\displaystyle \textup{Perimeter}=2(a+b)\), where \(\displaystyle a\) and \(\displaystyle b\) are the measurements of adjacent sides. 

Thus, the solution is:

\(\displaystyle 80=2(24+b)\)

\(\displaystyle 80=48+2b\)

\(\displaystyle 2b=80-48=32\)

\(\displaystyle b=\frac{32}{2}=16\)

A parallelogram must have two sets of congruent/parallel opposite sides. This parallelogram must have two sides with a measurement of \(\displaystyle 25\) and two base sides each with a length of \(\displaystyle 16\). In this question, you are given the information that the parallelogram has a base of \(\displaystyle 16\) and a total perimeter of \(\displaystyle 82\). Thus, work backwards using the perimeter formula in order to find the sum of the two adjacent sides to the base.

\(\displaystyle \textup{Perimeter}=2(a+b)\), where \(\displaystyle a\) and \(\displaystyle b\) are the measurements of adjacent sides. 

Thus, the solution is:

\(\displaystyle 82=2(16+b)\)

\(\displaystyle 82=32+2b\)

\(\displaystyle 2b=82-32=50\)

A parallelogram must have two sets of congruent/parallel opposite sides. Therefore, this parallelogram must have two sides with a measurement of \(\displaystyle 45\textup{mm}\) and two base sides each with a length of \(\displaystyle 55\textup{mm}\). To find the perimeter of the parallelogram apply the formula: 

\(\displaystyle \textup{Perimeter}=2(a+b)\), where \(\displaystyle a\) and \(\displaystyle b\) are the measurements of adjacent sides. 

Thus, the solution is:

\(\displaystyle p=2(55+45)\)

\(\displaystyle p=2(100)\)

\(\displaystyle p=200\)

A parallelogram must have two sets of congruent/parallel opposite sides. Therefore, this parallelogram must have two sides with a measurement of \(\displaystyle 136\textup{ inches}\) and two base sides each with a length of \(\displaystyle 140\textup{ inches}\). However, to solve this problem you must first convert the provided perimeter measurement from feet to inches. Since an inch is \(\displaystyle \frac{1}{12}\) of \(\displaystyle 1\) foot, \(\displaystyle 46\) feet is equal to\(\displaystyle 46\times12=552\) inches.

Now, you can work backwards using the formula: 

\(\displaystyle \textup{Perimeter}=2(a+b)\), where \(\displaystyle a\) and \(\displaystyle b\) are the measurements of adjacent sides. 

Thus, the solution is:

\(\displaystyle 552=2(140+b)\)

\(\displaystyle 552=280+2b\)

\(\displaystyle 2b=552-280=272\) 

\(\displaystyle b=\frac{272}{2}=136\)

A parallelogram must have two sets of congruent/parallel opposite sides. Therefore, this parallelogram must have two sides with a measurement of \(\displaystyle 16\) and two base sides each with a length of \(\displaystyle 18.\) Since the perimeter and one base length is provided in the question, work backwards using the perimeter formula:

\(\displaystyle \textup{Perimeter}=2(a+b)\), where \(\displaystyle a\) and \(\displaystyle b\) are the measurements of adjacent sides. 

Thus, the solution is:

\(\displaystyle 68=2(18+b)\)

\(\displaystyle 68=36+2b\)

\(\displaystyle 2b=68-36=32\)

\(\displaystyle b=\frac{32}{2}=16\)

A parallelogram must have two sets of congruent/parallel opposite sides. This parallelogram must have two sides with a measurement of \(\displaystyle 14\) and two base sides each with a length of \(\displaystyle 22\). In this question, you are provided with the information that the parallelogram has a base of \(\displaystyle 22\) and a total perimeter of \(\displaystyle 72\). Thus, work backwards using the perimeter formula in order to find the sum of the two adjacent sides to the base.

\(\displaystyle \textup{Perimeter}=2(a+b)\), where \(\displaystyle a\) and \(\displaystyle b\) are the measurements of adjacent sides. 

Thus, the solution is:

\(\displaystyle 72=2(22+b)\)

\(\displaystyle 72=44+2b\)

\(\displaystyle 2b=72-44=28\)