ACT Math Skill Review: Quadrilaterals

The ACT will test your knowledge of several types of four-sided figures (quadrilaterals). Indeed, you will be expected to solve many types of questions utilizing the information these figures offer. The most common four-sided figures are the rectangle and the square, followed by the parallelogram and the trapezoid. Note: for any four-sided (or three-sided, or more-sided) figure, the perimeter is the sum of the lengths of all the sides.

Let’s take a look at the most common types of quadrilaterals, starting with the most general, the parallelogram.

The Parallelogram

Parallelogram Facts:

• A parallelogram is a four-sided figure.
• Both pairs of opposite sides are parallel and equal.
• Both pairs of opposite angles are equal.
• The area of a parallelogram is its base times its height (Area of a parallelogram = b ∙ h), but due to its shape, the height of the parallelogram is not always equal to one of its sides. 
• To find the height of a parallelogram, you must draw a perpendicular line from the base to the top of the figure.

The Rectangle

Rectangle Facts 

• A rectangle is a parallelogram.
• All four interior angles are equal to 90°. 
• Opposite sides of a rectangle are equal.
• The diagonal of a rectangle makes two equal right triangles.
• The Pythagorean Theorem can be used to figure out details regarding the length of sides or the diagonal, depending on what information you’re given.
• The area of a rectangle is its length times its width (Area of a rectangle = l ∙ w).
• The perimeter of a rectangle is the sum of the lengths of its sides, or twice its length plus twice its width (2l + 2w)

The Square

Square Facts:

• A square is a type of rectangle whose four sides are equal in length.
• The diagonal of a square makes two 45°-45°-90° triangles with the sides of the square.
• You can figure out the length of the sides from the length of the diagonal and vice versa. 
• The area of a square is equal to the square of its side (Area of a square = s²). (Since the square is a rectangle, the area of a square is also length times width. However, since each side (s) is the same, the area of a square is usually represented as s².)
• The perimeter of a square is the sum of the lengths of its sides, or four times the length of its side (4s).

Trapezoids

Trapezoid Facts

• A trapezoid is a quadrilateral that only has one pair of parallel sides.
• The easiest way to find the area of a trapezoid is to divide it into two triangles and a rectangle, figure out the areas of the individual pieces and add the results together to find the area of the whole figure.

Examples

Answers and Explanations

1. The correct answer is B. If a question describes a shape but doesn’t provide you with a picture, it is usually helpful to draw it out. Here’s an example of what you’d end up with (note: it doesn’t have to be exact so don’t belabor it!).

   

   If we recall our parallelogram facts, opposite interior angles are equal. Thus, angle C is also 35°. Drawing it out helps to see that angles A and C are opposite. Angles B and D are also equal. Since all the angles add up to 360°,  B + D must equal 360 – (35 + 35). Thus, B + D = 290 and B and D both equal 145°. Notice that 145° and 290° both make an appearance in the answer choices as well, making it easy to make a careless mistake on a relatively straightforward problem.

2. The correct answer is H. In any quadrilateral, the interior angles must add up to 360°. We have values for two of the angles (19° and 89°) which together total 108°. Thus, there are 360° – 108°, or 252°  unaccounted for. These 252°  can be divided many ways, as long as the sum remains unchanged. The only answer choice that adds up to 252 is choice H.