Polygons, including Quadrilaterals

Introduction to Polygons

As we saw in the Two- and Three-Dimensional Figures section, polygons are coplanar closed shapes that are formed by straight line segments, which are called sides. Common polygons include rectangles, squares, and triangles. Circles, for example, are not polygons, since they are not formed by straight lines.

A polygon can be convex, where no points of any of the lines are in the interior of the shape, or concave, where some of the lines are in the interior. I like to identify concave polygons by looking for a “cave” or indentation in the shape.

Here is an example of a concave polygon, which has $ 5$ sides (pentagon). See how there’s a “cave” or indentation on the left-hand side? . Here is an example of a regular convex pentagon; notice any line segment between any two points on the figure would be either inside the figure or on the boundary: .

Note that this polygon has $ 5$ congruent sides and $ 5$ congruent angles; it is both equilateral, meaning that all sides are the same lengths, and equiangular, meaning that all angles are congruent. Thus, it is a regular polygon.

A few more definitions: The vertices of polygons are the endpoints of the sides, and diagonals of polygons are the segments that connect two non-consecutive (non-touching) sides, such as the diagonals of a rectangle.

Here are the names of polygons. Notice also that a larger number of sides is just called a “$ n$”-gon. For example, a polygon with $ 15$ sides is a $ 15$-gon.

Number of Sides	Polygon	Number of Sides	Polygon
$ 3$	Triangle	$ 8$	Octagon
$ 4$	Quadrilateral	$ 9$	Nonagon
$ 5$	Pentagon	$ 10$	Decagon
$ 6$	Hexagon	$ 11$	Hendecagon
$ 7$	Heptagon	$ 12$	Dodecagon

Angles of Polygons

There two main theorems having to do the sums of angles of polygons:

The Exterior Angle Sum states that the sum of the exterior angles of an $ n$-sided convex polygon is always $360{}^\circ $. For regular polygons (all sides and angles equal), each exterior angle of the polygon measures $ \displaystyle S=\frac{{360}}{n}$, where $ n$ is the number of sides.
The Interior Angle Sum states that the sum of the interior angles of an $ n$-sided convex polygon, one angle at each vertex, is $ \displaystyle S=\left( {n-2} \right)\times 180$, where $ n$ is the number of sides. (This seems weird, but this is because an $ n$-gon polygon can be divided up into $ n-2$ triangles, each with angles totaling $ 180{}^\circ$). For regular polygons (all sides and angles equal), each interior angle of the polygon measures $ \displaystyle S=\frac{{\left( {n-2} \right)\times 180}}{n}$.

For example, for a regular pentagon (five-sided figure):

Sum of exterior angles: $ 360{}^\circ$; each exterior angle: $ \displaystyle \frac{{360}}{5}=72{}^\circ$;
Sum of interior angles: $ \displaystyle \left( {5-2} \right)\times 180=540{}^\circ$; each interior angle: $ \displaystyle \frac{{\left( {5-2} \right)\times 180}}{5}=108{}^\circ$.

Quadrilaterals

A quadrilateral is a $ 4$-sided polygon; they have several unique characteristics. Examples of quadrilaterals include parallelograms (including rectangles and squares), kites, and trapezoids.

Here is a Venn Diagram (diagram with overlapping sections) that shows the different relationships among different types of quadrilaterals:

Parallelograms

Quadrilaterals with both sets of opposite sides parallel are (duh!) named parallelograms. Each pair of opposite, parallel sides is called a pair of bases, and any perpendicular segment between a pair of bases is called the altitude. The symbol $ \unicode{x25B1}$ is used to designate a parallelogram. (Note that the area of a parallelogram is, like a rectangle, $ A=Bh$, where the base is perpendicular to the height, or altitude.)

Note that all rectangles are parallelograms!

Here are some important properties of parallelograms:

Opposite sides are congruent.
Opposite angles are congruent.
Consecutive angles (angles next to each other) are supplementary, meaning their sum is $180{}^\circ$.
If there is one right angle, all four angles are right angles (rectangle).
The diagonals bisect each other.
Each diagonal divides the parallelogram into two congruent triangles.

There are certain ways to prove that figures are parallelogram:

Both pairs of opposite sides are parallel (definition of a parallelogram).
Both pairs of opposite sides are congruent.
Both pairs of opposite angles are congruent.
The diagonals bisect each other.
One pair of opposite sides is congruent and parallel (be careful; a figure is not necessarily a parallelogram if one pair of opposite sides is congruent, and the other pair is parallel, as this polygon could be a trapezoid – see below).

Parallelograms on the Coordinate Plane

Using the Coordinate Plane and the rules above, the slope, distance, and midpoint formulas can be used to determine if a quadrilateral is a parallelogram.

For example, let’s determine whether $ ABCD$, with vertices are $ \displaystyle A:\left( {1,5} \right)$, $ \displaystyle B:\left( {4,5} \right)$, $ \displaystyle C:\left( {0,1} \right)$, and $ \displaystyle D:\left( {3,1} \right)$, is a parallelogram. You can see that it probably is, but let’s “prove” it:

We can use either the Slope Method, or Distance Formula Method:

Slope Method: The slope of a line containing points $ \left( {{{x}_{1}},{{y}_{1}}} \right)$ and $ \left( {{{x}_{2}},{{y}_{2}}} \right)$ is $ \displaystyle m=\frac{{{{y}_{2}}-{{y}_{1}}}}{{{{x}_{2}}-{{x}_{1}}}}$.

Thus, $ \displaystyle \overline{{AB}}\parallel \,\overline{{CD}}$ (each have a slope of $ 0$) and $ \displaystyle \overline{{AC}}\parallel \,\overline{{BD}}$ (each have a slope of $ 4$). Since opposite sides have the same slope, they are parallel. Thus, by definition, $ ABCD$ is a parallelogram.

Distance Formula: The distance between two points $ \left( {{{x}_{1}},{{y}_{1}}} \right)$ and $ \left( {{{x}_{2}},{{y}_{2}}} \right)$ is $ d=\sqrt{{{{{\left( {{{x}_{2}}-{{x}_{1}}} \right)}}^{2}}+{{{\left( {{{y}_{2}}-{{y}_{1}}} \right)}}^{2}}}}$.

$ \text{AB}=\sqrt{{{{{\left( {{4}-{1}} \right)}}^{2}}+{{{\left( {{5}-{5}} \right)}}^{2}}}}=3$, $ \text{CD}=\sqrt{{{{{\left( {{3}-{0}} \right)}}^{2}}+{{{\left( {{1}-{1}} \right)}}^{2}}}}=3$, $ \text{AC}=\sqrt{{{{{\left( {{0}-{1}} \right)}}^{2}}+{{{\left( {{1}-{5}} \right)}}^{2}}}}= \sqrt{{17}}$, $ \text{BD}=\sqrt{{{{{\left( {{3}-{4}} \right)}}^{2}}+{{{\left( {{1}-{5}} \right)}}^{2}}}}= \sqrt{{17}}$.

Thus, $ \displaystyle \overline{{AB}}\cong \,\overline{{CD}}$ and $ \displaystyle \overline{{AC}}\cong \,\overline{{BD}}$. Since opposite sides are congruent to each other, $ ABCD$ is a parallelogram.

Characteristics of Special Quadrilaterals

Here is a table of some special types of quadrilaterals and their characteristics, as shown in the Venn Diagram above.

Rectangle	Rhombus	Trapezoid	Kite
Parallelogram (Quadrilateral) with four right angles. Diagonals are congruent. Squares are rectangles with four congruent sides. All squares are rectangles, but not all rectangles are squares.	Parallelogram with four congruent sides. Diagonals are perpendicular (form four right angles). All squares are rhombuses (or rhombi), but not all rhombuses are squares.	Quadrilateral with exactly one pair of parallel sides (called bases). The midsegment or median is the segment that connects the midpoints of the legs. From the Midpoint Theorem, this length is equal to $ \displaystyle \frac{1}{2}$ the sum of the bases. Isosceles trapezoids have congruent legs, and both pairs of base angles are congruent, and diagonals are congruent:	Quadrilateral with exactly two pairs of consecutive congruent sides. Note that opposite sides of a kite are not congruent or parallel. Diagonals are perpendicular (form four right angles). Only one pair of opposite angles is congruent.

Rectangle

Rhombus

Trapezoid

Kite

Parallelogram (Quadrilateral) with four right angles.
Diagonals are congruent.
Squares are rectangles with four congruent sides. All squares are rectangles, but not all rectangles are squares.

Parallelogram with four congruent sides.
Diagonals are perpendicular (form four right angles).
All squares are rhombuses (or rhombi), but not all rhombuses are squares.

Quadrilateral with exactly one pair of parallel sides (called bases).
The midsegment or median is the segment that connects the midpoints of the legs. From the Midpoint Theorem, this length is equal to $ \displaystyle \frac{1}{2}$ the sum of the bases.
Isosceles trapezoids have congruent legs, and both pairs of
base angles are congruent, and diagonals are congruent:

Quadrilateral with exactly two pairs of consecutive congruent sides. Note that opposite sides of a kite are not congruent or parallel.
Diagonals are perpendicular (form four right angles).
Only one pair of opposite angles is congruent.

Problems and Solutions

Problem: One interior angle of a regular polygon measures $ 140{}^\circ$. What is the polygon?

Solution: Use the Interior Angle Sum theorem for polygons, where $ n$ is the number of sides: $ \displaystyle \frac{{\left( {n-2} \right)\times 180}}{n}=140;\,\,\,\left( {n-2} \right)\times 180=140n;\,\,\,180n-360=140n;\,\,40n=360;\,\,n=9$. The polygon is a nonagon (nine sides).

Problem: One exterior angle of a regular polygon measures $ 15{}^\circ$. What is the polygon?

Solution: Use the Exterior Angle Sum theorem for polygons: $ \displaystyle \frac{{360}}{n}=15;\,\,15n=360;\,\,n=24$. The polygon has $ 24$ sides.

Problems and Solutions: True or False?

A square is a rhombus. | True; a square is a quadrilateral with all sides congruent.
A quadrilateral with one right angle is a rectangle. | False; it could be a rectangle, but could also be a trapezoid.
A parallelogram with one right angle is a rectangle. | True; since a parallelogram has congruent opposite sides, and congruent opposite angles, it must be a rectangle.
A trapezoid is a quadrilateral. | True, by definition.
The diagonals of an isosceles trapezoid bisect each other. | False; the diagonals are equal in length, but do not bisect each other.
All quadrilaterals are either trapezoids or parallelograms. | False; they could be kites, for example.
If the diagonals of a rhombus are perpendicular, the rhombus is a square. | False; all rhombi have perpendicular diagonals, even those that aren’t squares.
If a quadrilateral has opposite sides parallel, consecutive sides congruent, and consecutive angles congruent, it is a square. | True; its angles must be right angles, and its sides all have the same measurement.
If the diagonals of a parallelogram are perpendicular, it is a rectangle. | False; it wouldn’t necessarily have to be a rectangle, but it must be a rhombus.
If two consecutive angles of a quadrilateral are right angles, then it is a rectangle. | False; it could be a trapezoid.
A quadrilateral has a pair of opposite congruent sides as well as a pair of opposite parallel sides. The quadrilateral is a parallelogram. | False; it could be an isosceles trapezoid: . Note that if the same sides were both congruent and parallel, it would be a parallelogram.

Problem: Find $ x$ and $ y$ so that the figure is a parallelogram.

Solution: Since the figure is a parallelogram, alternate interior angles must be congruent. Thus, $ 4x=20; x=5$. Also, $ x+2y=25; 5+2y=25; y=10$.

Problem: Is the quadrilateral $ ABCD$ (vertices $ \displaystyle \left( {1,5} \right)$, $ \displaystyle \left( {2,7} \right)$, $ \displaystyle \left( {4,6} \right)$, and $ \displaystyle \left( {3,4} \right)$) a rectangle, square, or rhombus? Show work.

Solution: Using the Distance Formula, $ \text{AB}=\sqrt{{{{{\left( {{2}-{1}} \right)}}^{2}}+{{{\left( {{7}-{5}} \right)}}^{2}}}}=\sqrt{5}$, $ \text{BC}=\sqrt{{{{{\left( {{4}-{2}} \right)}}^{2}}+{{{\left( {{6}-{7}} \right)}}^{2}}}}=\sqrt{5}$, $ \text{CD}=\sqrt{{{{{\left( {{3}-{4}} \right)}}^{2}}+{{{\left( {{4}-{6}} \right)}}^{2}}}}=\sqrt{5}$, and $ \text{DA}=\sqrt{{{{{\left( {{1}-{3}} \right)}}^{2}}+{{{\left( {{5}-{4}} \right)}}^{2}}}}=\sqrt{5}$. The slopes of $ AB$ and $ DC$ are the same ($ 2$), and the slopes of $ BC$ and $ AD$ are the same ($ \displaystyle -\frac{1}{2}$). Since the two sets of lines have negative reciprocal slopes, they are perpendicular.

Since all sides are the same length, opposite sides have the same slope, and consecutive sides are perpendicular, the quadrilateral is a square, a rectangular, and a rhombus.

Problem: Prove the following using a two-column proof:

Given: $ \overline{{AC}}$ and $ \overline{{EF}}$ bisect each other, and $ \displaystyle \angle \,1=\angle \,2$.

Prove: $ ABCD$ is a $ \unicode{x25B1}$ (parallelogram).

Solution:

Proof:

Statements	Reasons
1. $\overline{{AC}}$ and $ \overline{{EF}}$ bisect each other.	1. Given
2. $ AG=GC$, $ EG=GF$	2. Def of Bisector
3. $ \displaystyle \angle \,5\cong \angle \,6$	3. Vertical Angles
4. $ \displaystyle \vartriangle AGE\cong \vartriangle CGF$	4. SAS Triangle Congruency
5. $ \displaystyle \angle \,4\cong \angle \,7$	5. CPCTC
6. $ \displaystyle \overline{{AB}}\parallel \overline{{DC}}$	6. AIA (Alternate Interior Angle) $\angle \text{ }\!\!’\!\!\text{ s}\,\,\,\text{are}\,\cong \displaystyle \,\,\,\Rightarrow $ lines are $ \parallel$
7. $ \displaystyle \angle \,1\cong \angle \,2$	7. Given
8. $ \displaystyle \overline{{AD}}\parallel \overline{{BC}}$	8. AIA (Alternate Interior Angle) $\angle \text{ }\!\!’\!\!\text{ s}\,\,\,\text{are}\,\cong \displaystyle \,\,\,\Rightarrow $ lines are $ \parallel$
9. $ ABCD$ is a $ \unicode{x25B1}$ (parallelogram).	9. Definition of parallelogram

Problem: Given $ ABCD$ is a kite, find the measurement of $ BC$.

Solution: Since the diagonals of a kite are perpendicular, and $ AC$ and $ DB$ bisect each other, use the Pythagorean Theorem to find $ BC$: $ \displaystyle {{8}^{2}}+{{15}^{2}}=B{{C}^{2}};\,\,\,\,BC=\sqrt{{289}}=17$.