Since \(\displaystyle \angle 1\) is acute, a right triangle can be constructed with an altitude as one leg and a side as the hypotenuse, as is shown here. The height of the triangle must be less than its sidelength of 8 inches.

The height of the parallelogram must be less than its sidelength of 8 inches.

The area of the parallelogram is the product of the base and the height - which is \(\displaystyle A = 15 \cdot n\)

 Therefore, 

\(\displaystyle n < 8\)

\(\displaystyle 15 \cdot n < 15 \cdot 8\)

\(\displaystyle A < 120\)

(B) is greater.

The area of a parallelogram is the product of its height and its base; its slant length is irrelevant. Both parallelograms have the same height (8 inches) and the same base (1 foot, or 12 inches), so they have the same areas.

A rhombus, by definition, has four sides of equal length. Therefore, \(\displaystyle AB= CD\). Also, since \(\displaystyle X\) and \(\displaystyle Y\) are the midpoints of their respective sides, 

\(\displaystyle AX = XB = \frac{1}{2 }AB=\frac{1}{2 } CD = CY = YD\)

We will assign \(\displaystyle n\) to the common length of the four half-sides of the rhombus.

Also, both \(\displaystyle \overline{AY}\) and \(\displaystyle \overline{XC}\) are altitudes of the rhombus; the are congruent, and we will call their common length \(\displaystyle h\) (height).

The figure, with the lengths, is below.

Rectangle \(\displaystyle AXCY\) has dimensions \(\displaystyle h\) and \(\displaystyle n\); its area, 150, is the product of these dimensions, so

\(\displaystyle A_{1} = hn = 150\)

The area of the entire Rhombus \(\displaystyle ABCD\) is the product of its height \(\displaystyle h\) and the length of a base \(\displaystyle 2n\), so

\(\displaystyle A = h \cdot 2n = 2hn = 2 \cdot hn = 2 \cdot 150 = 300\).

The measure of \(\displaystyle \angle 1\) is actually irrelevant. The perimeter of the parallelogram is the sum of its four sides; since opposite sides of a parallelogram have the same length, the perimeter is

\(\displaystyle P = 8 + 15 + 8 + 15 = 46\) inches, 

making the quantities equal.

The perimeter of a parallelogram is the sum of its sidelengths; its height is irrelevant. Also, opposite sides of a parallelogram are congruent.

The perimeter of parallelogram A is 

\(\displaystyle 12 + 12 + 12 + 12 = 48\) inches;

The perimeter of parallelogram B is 

\(\displaystyle 12 + 10 + 12 + 10 = 46\) inches.

(A) is greater.

All four sides of a rhombus have the same length, so we can find the perimeter of Rhombus \(\displaystyle ABCD\) by taking the length of one side and multiplying it by four. Since \(\displaystyle AB = 32\), the perimeter is four times this, or \(\displaystyle 4 \cdot AB = 4 \cdot 32 = 128\).

Note that the length of \(\displaystyle \overline{AY}\) is actually irrelevant to the problem.

In Parallelogram \(\displaystyle ABCD\), \(\displaystyle \overline{AB}\) and \(\displaystyle \overline{BC }\) are adjoining sides; there is no specific rule for the relationship between their lengths. Therefore, no conclusion can be drawn of \(\displaystyle AB\) and \(\displaystyle BC\), and no conclusion can be drawn of the relationship between \(\displaystyle x\) and \(\displaystyle y\).

Opposite angles of a parallelogram must have the same measure, so the correct choice must have two pairs, each of the same angle measure. We can therefore eliminate \(\displaystyle 38^{\circ }, 142 ^{\circ }, 28^{\circ }, 152 ^{\circ }\) and \(\displaystyle 102^{\circ }, 94 ^{\circ }, 86^{\circ }, 78 ^{\circ }\) as choices. 

Also, the sum of the measures of the angles of any quadrilateral must be \(\displaystyle 360 ^{\circ }\), so we add the angle measures of the remaining choices:

\(\displaystyle 88^{\circ }, 97 ^{\circ }, 88^{\circ }, 97 ^{\circ }\): 

\(\displaystyle 88^{\circ } + 97 ^{\circ } + 88^{\circ }+ 97 ^{\circ } = 370\), so we can eliminate this choice.

\(\displaystyle 77^{\circ }, 133 ^{\circ }, 77^{\circ }, 133 ^{\circ }\):

\(\displaystyle 77^{\circ } + 133 ^{\circ } + 77^{\circ }+ 133 ^{\circ } =420\), so we can eliminate this choice.

\(\displaystyle 57^{\circ }, 123 ^{\circ }, 57^{\circ }, 123 ^{\circ }\)

\(\displaystyle 57^{\circ } +123 ^{\circ }+57^{\circ }+123 ^{\circ } = 360\); this is the correct choice.

The sum of two consecutive angles of a parallelogram is \(\displaystyle 180^{\circ }\).

\(\displaystyle (x+23) + y = 180\)

\(\displaystyle x + y +23= 180\)

\(\displaystyle x + y +23 -23= 180-23\)

\(\displaystyle x + y = 157\)

157 is the correct choice.

In Parallelogram \(\displaystyle ABCD\), \(\displaystyle \angle A\) and \(\displaystyle \angle C\) are opposite angles and are therefore congruent. This means that

\(\displaystyle m \angle A = m \angle C\)

\(\displaystyle 3x= 2y\)

\(\displaystyle 3x \div 3 = 2y \div 3\)

\(\displaystyle x = \frac{2}{3}y\)

Both are positive, so \(\displaystyle x< y\).