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
4 sides
Quadrilateral
5 sides
Pentagon
6 sides
Hexagon
8 sides
Octagon
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
Figure 6.2.1: 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
Isosceles
Only two sides are equal
Scalene
All three sides are different in length
ANGLES
Acute
Each angle is less than 900
Right
One angle is 900
Obtuse
One angle is more than 900
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?
Polygon
Quadrilateral
Parallelogram
Trapezoid
Figure 6.2.2
What is the difference between a regular and irregular polygon?
Facts about Angles
Figure 6.2.3
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.
Figure 6.2.4
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.
Figure 6.2.5: Complementary angles
Definition: Supplementary Angles
Supplementary angles are any two angles with a sum of 1800. See angles A and B below.
Figure 6.2.6: Supplementary angles
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?
Practice Problems
(Problems 1 – 4) Find the measure of angle b.
(Problems 5 – 6) Find the measure of each angle indicated.
(Problems 7 – 10) Classify each angle as acute, obtuse, right or straight.
\(121^{\circ}\)
\(180^{\circ}\)
(Problems 11 – 12) Classify each triangle by its angles.
(Problems 13 – 14) Classify each triangle by its angles and sides.
(Problems 15 – 16) Sketch an example of the type of triangle described.
Acute Isosceles
Right Obtuse
(Problems 17 – 18) Write the name of each polygon.
(Problems 19 – 22) Find the interior angle sum for each polygon. Round your answer to the nearest tenth, if necessary.
(Problems 23 – 26) State if the polygon is regular or irregular.