Lesson 8.2: Quadrilaterals and Polygons

Introduction

In this lesson, we will explore the fascinating world of quadrilaterals and polygons, crucial topics in GRE General’s Quantitative Reasoning segment. Our objectives will focus on understanding the properties of various quadrilaterals like rectangles, squares, parallelograms, and trapezoids. We will also delve into the concepts of interior and exterior angles of polygons, alongside calculating area and perimeter for these shapes. By the end of this lesson, students will be equipped to handle GRE-style questions with confidence.

Learning Objectives:

Properties of rectangles, squares, parallelograms, and trapezoids.
Interior and exterior angles of polygons.
Area and perimeter of polygonal figures.
Apply properties of common quadrilaterals.
Compute interior and exterior angle measures.

Properties of Quadrilaterals

Quadrilaterals are four-sided figures that come in various forms, each defined by its properties. Let’s discuss the most common types of quadrilaterals: rectangles, squares, parallelograms, and trapezoids.

Rectangles

A rectangle is a quadrilateral with opposite sides equal in length and four right angles. The properties of rectangles can be summarized as follows:

Opposite sides are equal: If $ABCD$ is a rectangle, then $AB = CD$ and $BC = AD$.
All angles are right angles: $\angle A = \angle B = \angle C = \angle D = 90^\circ$.
Diagonals are equal: The diagonals $AC$ and $BD$ are equal in length and bisect each other.

Example 1: Area and Perimeter of a Rectangle

Suppose we have a rectangle with a length of 8 units and a width of 5 units.

Area is calculated using the formula:

$$ A = \text{Length} \times \text{Width} $$

$$ A = 8 \times 5 = 40 \text{ square units} $$

Perimeter is calculated as:

$$ P = 2 \times (\text{Length} + \text{Width}) $$

$$ P = 2 \times (8 + 5) = 26 \text{ units} $$

Squares

A square is a special type of rectangle where all four sides are equal, and all angles are right angles. The properties of squares include:

All sides are equal: If $ABCD$ is a square, then $AB = BC = CD = DA$.
All angles are right angles: $\angle A = \angle B = \angle C = \angle D = 90^\circ$.
Diagonals are equal and bisect at 90 degrees.

Example 2: Area and Perimeter of a Square

Consider a square with each side measuring 6 units.

Area:

$$ A = \text{Side}^2 $$

$$ A = 6^2 = 36 \text{ square units} $$

Perimeter:

$$ P = 4 \times \text{Side} $$

$$ P = 4 \times 6 = 24 \text{ units} $$

Parallelograms

A parallelogram is defined by having opposite sides that are both equal and parallel. The properties include:

Opposite sides are equal: If $ABCD$ is a parallelogram, then $AB = CD$ and $BC = AD$.
Opposite angles are equal: $\angle A = \angle C$ and $\angle B = \angle D$.
Adjacent angles are supplementary: $\angle A + \angle B = 180^\circ$.

Example 3: Area of a Parallelogram

If we have a parallelogram with a base of 10 units and a height of 4 units:

Area:

$$ A = \text{Base} \times \text{Height} $$

$$ A = 10 \times 4 = 40 \text{ square units} $$

Trapezoids

A trapezoid (or trapezium) is a quadrilateral with at least one pair of parallel sides. The properties of trapezoids include:

One pair of opposite sides is parallel (the bases).
The non-parallel sides are called the legs.

For an isosceles trapezoid, the legs are equal, and the angles adjacent to each base are equal.

Example 4: Area of a Trapezoid

Consider a trapezoid with bases of 6 units and 10 units, and a height of 5 units.

Area:

$$ A = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height} $$

$$ A = \frac{1}{2} \times (6 + 10) \times 5 = 40 \text{ square units} $$

Interior and Exterior Angles of Polygons

Polygons are closed figures with three or more sides. The sum of the interior angles of a polygon is determined by the formula:

$$ S = (n - 2) \times 180^\circ $$

where $n$ is the number of sides.

The exterior angle of a polygon is formed by one side of the polygon and the extension of an adjacent side. The sum of the exterior angles of any polygon is always $360^\circ$, regardless of the number of sides.

Example 5: Calculating Interior Angles

Consider a hexagon (6-sided polygon). The sum of the interior angles is:

$$ S = (6 - 2) \times 180^\circ = 720^\circ $$

If the hexagon is regular (all angles are equal), each interior angle is:

$$ \text{Each angle} = \frac{720^\circ}{6} = 120^\circ $$

Example 6: Exterior Angles of a Polygon

For the same hexagon, the sum of the exterior angles is always:

$$ 360^\circ $$

If the hexagon is regular, each exterior angle is:

$$ \text{Each angle} = \frac{360^\circ}{6} = 60^\circ $$

Conclusion

In this lesson, we covered the essential properties of quadrilaterals and polygons. students should now be able to identify and demonstrate knowledge of rectangles, squares, parallelograms, and trapezoids, as well as compute both area and perimeter. Do not forget the crucial properties of interior and exterior angles in polygons. Mastering these concepts will greatly assist in solving related GRE problems efficiently.

Study Notes

Quadrilaterals are defined as four-sided polygons, including rectangles, squares, parallelograms, and trapezoids.
The area and perimeter formulas differ by the type of quadrilateral.
The sum of interior angles of a polygon is calculated using $S = (n - 2) \times 180^\circ$.
The sum of the exterior angles of a polygon is always $360^\circ$.
Regular polygons have equal sides and angles, leading to consistent angle measures.