Section 9'2: Polygons

1
Section 9.2 Polygons

• Dr. Fred Butler
• Math 121 Fall 2004

2
Polygons

• A polygon is a closed figure in a plane
  determined by three or more straight line
  segments.
• Below are pictured some examples of polygons.

3
Polygons contd.

• The straight line segments that form the polygon
  are called its sides, and a point where two sides
  meet is called a vertex.
• The union of the sides of a polygon and its
  interior is called a polygonal region.
• A regular polygon is one whose sides are all the
  same length, and whose interior angles have the
  same measure.
• The triangle and square pictured are regular
  polygons.

4
Polygon Names
5
Sum of Angles in a Triangle

• In the picture below lines l1 and l2 are
  parallel, so ltAltA, ltBltB, and ltCltC and
• ltAltBltC 180.
• We see that the sum of the interior angles of the
  triangle pictured is also 180, that is
• ltAltBltC 180.

6
Sum of Angles in a Quadrilateral

• If we draw a line between two non-adjacent
  vertices in any quadrilateral, we see that we
  have two triangles.
• Since the sum of the interior angles of a
  triangle is 180 and any quadrilateral is made up
  of two triangles, we see that the sum of the
  interior angles of any quadrilateral is 2x 180
  360 .

7
Sum of Angles in Pentagon and Hexagon

• Using the same idea, we see that any pentagon is
  made up of three triangles, and any hexagon is
  made up of four triangles.
• Thus the sum of the interior angles of a pentagon
  is 3x 180 540, and the sum of the interior
  angles of a hexagon is 4x180 720.

8
Sum of Angles in n-Sided Polygon

• We see in general the sum of the measures of the
  interior angles of an n-sided polygon is
• (n 2)x180.

9
PRS Question 4.9

• What is the sum of the measures of the interior
  angles in a dodecagon?
• 1. 180 2. 2160 3. 1800

10
Types of Triangles
11
PRS Question 4.10

• Which of the following is the triangle pictured
  below NOT?
• 1. acute 2. right 3. isosceles

12
Similar Figures

• Figures that have the same shape but may be
  different sizes are called similar figures.
• An example of two similar figures is pictured
  below.

13
Similar Figures contd.

• Similar figures have corresponding angles and
  corresponding sides.
• For example, triangle ABC has angles A, B, and C
  which correspond in triangle DEF to angles D, E,
  and F respectively.
• Sides AB, BC, and AC in triangle ABC correspond
  respectively to sides DE, EF, and DF in triangle
  DEF.

14
Formal Definition of Similar Polygons

• Two polygons are similar if their corresponding
  angles have the same measure and their
  corresponding sides are in proportion.

15
An Example of Similar Polygons

• As we said earlier, triangles ABC and DEF are
  similar.
• Thus ltA and ltD have the same measure, as does ltB
  and ltE, and ltC and ltF.
• Also, the corresponding sides are in proportion
• AB/DE BC/EF AC/DF.

16
Using Similarity

• The figures to the right are similar. Let us
  determine the length of side CD.
• We will represent the length of side CD by the
  variable x.
• Since the figures are similar, the corresponding
  side lengths are in proportion.

17
Using Similarity contd.

• The corresponding side of CD is OP.
• Side lengths AE and MQ are known, so we can use
  them as one ratio in the proportion.
• Thus
• AE/MQ CD/OP
• 8/10x/15.

18
Using Similarity contd.

• The length of side CD, represented by x,
  satisfies the equation
• 8/10x/15.
• Now we solve for x.
• 1. Cross multiply (8)(15)(10)(x)
• 2. Simplify 12010x
• 3. Divide both sides by 10 12x.
• So the length of side CD is 12.

19
Congruent Figures

• If the corresponding sides of two similar figures
  are the same length, the figures are called
  congruent figures.
• Corresponding angles of congruent figures have
  the same measure, and the corresponding sides are
  equal in length.
• Two congruent figures coincide when placed one on
  top of the other.

20
Example of Congruent Figures

• Triangles ABC and DEF are congruent. Let us
  determine the length of side DF and the measure
  of ltACB.
• Side DF corresponds to side AC in the top
  triangle, so side DF has length 12.
• ltACB corresponds to ltDFE in the bottom triangle,
  so the measure of ltACB is 34

21
Types of Quadrialterals
22
How Quadrilaterals Are Related

• trapezoid
• parallelogram
• rectangle rhombus
• square

23
PRS Question 4.11

• Which of the following is the quadrilateral
  pictured below NOT?
• 1. trapezoid 2. parallelogram
• 3. rhombus 4. rectangle

24
Lecture Summary

• A polygon is a closed figure in a plane
  determined by three or more straight line
  segments.
• The sum of the measures of the interior angles of
  an n-sided polygon is (n 2)x180.
• We discussed acute, obtuse, right, isosceles,
  equilateral, and scalene triangles.

25
Lecture Summary contd.

• Two polygons are similar if their corresponding
  angles have the same measure and their
  corresponding sides are in proportion.
• Two similar figures whose corresponding sides are
  the same length, the figures are called congruent
  figures.
• We discussed the trapezoid, parallelogram,
  rectangle, rhombus, square.

26
Homework

• Do problems from Section 9.2 of the textbook.
• I will be in the IML lab today 330-430 and
  tomorrow 1030-1130 for help with the lab.
• The lab is due Monday Nov. 08 by 1100 PM Web CT
  time.