7.6: Polygons

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Amy Lagusker
College of the Canyons

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Below is a table of polygons. There are an infinite amount of polygons, but the following are the shapes taught in elementary school.

Table 6.2.1: Polygons
Number of Sides	Name	Irregular Polygon	Regular Polygon
3 sides	Triangle	$clipboard_e9815a53801e983b55ed322ef83a27dad.png$	$clipboard_ea3e5f4c273546fc93cec49e188606375.png$
4 sides	Quadrilateral	$clipboard_ef97751f54ca101b86965e14f8e4c15bb.png$	$clipboard_ebeeb85777811192d8b6255aedc21f1eb.png$
5 sides	Pentagon	$clipboard_e14a6c1a69337affb6f579748a00d0ddf.png$	$clipboard_e7b434027e06a8170f418c78a0b951a81.png$
6 sides	Hexagon	$clipboard_eb1e4720be5b0e7e0de346ef74604e55d.png$	$clipboard_ec1168c568a4121d01bf108d493bdedfe.png$
8 sides	Octagon	$clipboard_e2ba5312a8ea77437e7076e6bd84ad5c6.png$	$clipboard_e83c745f3390f704e3f952a32b851e0c8.png$

Definition: Regular Polygon

A shape whose sides have the same length and whose angles have the same measure.

Definition: Irregular Polygon

A shape whose sides differ in length or have angles of different measure.

Hierarchy of Polygons

Polygon Definitions

Definition: Kite

A quadrilateral with two consecutive sides having equal lengths and the other two sides also have equal lengths.

Definition: Trapezoid

A quadrilateral with at least one pair of opposite sides parallel.

Definition: Isosceles Trapezoid

A trapezoid with both angles next to one of the parallel sides having the same size.

Definition: Parallelogram

A trapezoid with pairs of opposite sides parallel.

Definition: Rectangle

A parallelogram with a right angle.

Definition: Rhombus

A quadrilateral with all sides being the same.

Definition: Square

A rectangle that has four equal sides.

Types of Triangles

Table 6.2.2: Triangles
Name	Definition	Triangle
SIDES
Equilateral	All three sides are equal	$clipboard_e3df189067db6f2f13bdb92f43db4f66a.png$
Isosceles	Only two sides are equal	$clipboard_e333f30e222860718af08ce594b4bccea.png$
Scalene	All three sides are different in length	$clipboard_ea6fa1bf332f06943b4af87ea34fea2c5.png$
ANGLES
Acute	Each angle is less than 900	$clipboard_e333f30e222860718af08ce594b4bccea.png$
Right	One angle is 900	$clipboard_e8e1e31e785bcc8132a8048ae9b7615d0.png$
Obtuse	One angle is more than 900	$clipboard_e54ae0309e1700723b412133434271e2c.png$

Partner Activity 1

Draw the following triangles

Isosceles right triangle
Scalene obtuse triangle
Equilateral right triangle

Partner Activity 2

Is a rectangle a square? Is a square a rectangle?
Multiple Choice: Which one is NOT a name for the figure below?
1. Polygon
2. Quadrilateral
3. Parallelogram
4. Trapezoid

What is the difference between a regular and irregular polygon?

Facts about Angles

Angles in a triangle add up to 1800
An angle forming a straight line is also 1800
Any quadrilateral (4-sided figure) is 3600
Angles which round a point add up to 3600
The two base angles of an isosceles triangle are equal

Why does a triaangle add up to $180^{\circ }$

A full circle is $360^{\circ}$. Half of a circle, called a semicircle, would then be $180^{\circ }$. The diameter (a line which passes through the center of the circle) of the semicircle is then also $180^{\circ }$. Therefore, all straight lines are $180^{\circ }$. See the figure below. Knowing that all straight lines are $180^{\circ }$, we look at the figure below of the line and triangle.

Since a line is $180^{\circ }$, we know that angles $A_1$, B, and $C_1$ must add up to $180^{\circ }$. A theorem (proven statement) in Geometry states that alternate (opposite sides) interior angles are congruent (equal). Angles $A_1$ and $A_2$ are alternate interior, cut by the transversal (line) connecting angle $A_2$ to the straight line. Angles $C_1$ and $C_2$ follow a similar approach.
Since the measures of angles $A_{1}=A_{2}$, $C_{1}=C_{2}$, and $A_{1}+B+C_{1}=180$, then by substitution, $A_{2}+B+C_{2}=180$. Therefore, triangle $A_{2} B C_{2}$ adds up to $180^{\circ }$.

Partner Activity 3

The sum of the interior angles of any polygon is represented by: $180(n-2)$.

Find the sum of the interior angles of a triangle, using the formula.
Find the sum of the interior angles of a pentagon, using the formula.
Find the sum of the interior angles of a 15-sided polygon, using the formula.
What is the sum of the EXTERIOR angles of a pentagon?

Complementary and Supplementary Angles

Definition: Complementary Angles

Complementary angles are any two angles with a sum of 900. See angles C and D below.

Definition: Supplementary Angles

Supplementary angles are any two angles with a sum of 1800. See angles A and B below.

Partner Activity 4

You have two supplementary angles. One angle is 300. What is the measure of the other angle?
One angle is complementary to another angle. The first one is 490. What is the measure of the second angle?