Quadrilateral Word Problems Study Guide
Introduction to Quadrilateral Word Problems
[G]eometry is not true, it is advantageous.
—HENRI POINCARÉ (1854–1912)
This lesson will review the properties of quadrilaterals and the special types of quadrilaterals. Word problems involving quadrilaterals will be solved.
Quadrilaterals are four-sided polygons. This means that they are closed figures with four line segments as sides. The interior angles of every quadrilateral total 360°.
There are many types of special quadrilaterals. These quadrilaterals are classified, or named, based on their properties.
The first group of special quadrilaterals is the parallelograms.
A parallelogram is a quadrilateral with opposite sides congruent, or the same measure. Parallelograms also have opposite angles congruent, and the diagonals bisect each other. In addition, consecutive angles of any parallelogram are supplementary. In other words, angles that are next to each other, like angle A and angle B in the following figure, add to 180 degrees. This property and others are shown in the examples below.
A rhombus is a parallelogram with all four sides congruent. It could look like a square that is leaning over. Keep in mind that all squares are also rhombuses. Rhombuses have all the properties of parallelograms, in addition to the fact that the diagonals are perpendicular, or meet at right angles. Examples of rhombuses are shown in the following figures.
A rectangle is a parallelogram with four right angles. It could look like a parallelogram standing up straight. Rectangles also have all of the properties of parallelograms, in addition to the fact that the diagonals are congruent. Following are two examples of rectangles.
A square is a rhombus with right angles. Therefore, all sides are congruent, and all angles are right angles. Squares have all of the properties of rhombuses, in addition to having four congruent angles and congruent diagonals. A square is shown next.
A square can be remembered as a rectangle with four congruent sides, or a rhombus with four right angles.
The second group of special quadrilaterals is the trapezoids.
A trapezoid is a quadrilateral with exactly one pair of parallel sides. Trapezoids have two parallel sides that are not the same measure; these sides are called the bases of the trapezoid. The two sides that are not parallel are called the legs of the trapezoid. Trapezoid examples are shown in the following figures.
An isosceles trapezoid, like an isosceles triangle, has two sides congruent. In this type of trapezoid, the legs are the same measure. Isosceles triangles also have congruent base angles and congruent diagonals. These properties are shown in the following figure.
All parallelograms have two pairs of parallel sides.
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