1.5: Describe Angle Pair Relationships1.5: Describe Angle Pair Relationships1.6: Classify Polygons1.6: Classify Polygons

Objectives:

1.To use special angle relationships to find angle measures

2.To define, name, and classify polygons

Vocabulary Vocabulary (make sure you know (make sure you know these)these)

Complementary Supplementary Linear Pair

Vertical Angles Polygon Diagonal (n.)

Convex Concave Equilateral

Equiangular Regular Define these in Define these in your notebookyour notebook

C Comes Before S…C Comes Before S…

9043

9021

mm

mm

18087

18065

mm

mm

Example 1a Example 1a

1. Given that <1 is a complement of <2 and m<1 = 68°, find m<2.

2. Given that <3 is a supplement of <4 and m<3 = 56°, find m<4.

220

1240

Example 1b Example 1b

1. What is the sum of complementary angles in radians?

2. What is the sum of supplementary angles in radians?

3. What is complement for the angle that measures π/3?

4. What is the supplement for the angle that measures 3π/4?

Π 2

Π

Π6

Π 4

Example 2Example 2

Let <A and <B be complementary angles and let m<A = (2x2 + 35)° and m<B = (x + 10)°. What is (are) the value(s) of x? What are the measures of the angles?

Set up the equation and solve

X = 4.5 or -5m<A = 14.5 or 85m<B = 75.5 or 5Check to make sure the sum is 90

Linear Pairs of AnglesLinear Pairs of Angles

Linear Pairs of AnglesLinear Pairs of Angles

• Two adjacent angles form a linear pair linear pair if their noncommon sides are opposite rays.

• The angles in a linear pair are supplementarysupplementary.

Vertical AnglesVertical Angles

Vertical AnglesVertical Angles

• Two nonadjacent angles are vertical vertical anglesangles if their sides form two pairs of opposite rays.

• Vertical angles are formed by two intersecting lines.

Check them out HERE

Example 3Example 3

Identify all of the linear pairs of angles and all of the vertical angles in the figure.

Example 4: SATExample 4: SAT

In the figure and , what is the value of x?

5x

y4

x

z

zy

x

x=18, y=90 and z=72HOWHOW did I do that?

3-D Rendering3-D Rendering

3-D rendering in digital graphics is based upon polygons.

3-D Rendering3-D Rendering

The higher the polygon count, the smoother the surface.– Tomb Raider (1996)

3-D Rendering3-D Rendering

The higher the polygon count, the smoother the surface.– Tomb Raider

Underworld (2008)

What Makes a Polygon?What Makes a Polygon?

So, what makes a polygon a polygon?

PolygonsPolygons

A closed plane figure is a polygonpolygon if it is formed by 3 or more line segments (sidessides), joined endpoint to endpoint (verticesvertices) with each side intersecting exactly two others.

Parts of a PolygonParts of a Polygon

What’s the name of this polygon?

Consecutive Angles

Consecutive Vertices

Consecutive Sides

Example 5Example 5

Why are the following not polygons?

Names of PolygonsNames of Polygons (memorize (memorize these)these)

• Polygons come in many flavors.

• They are classified by the number of sides they have.

• A polygon with more than 12 sides is commonly called an n-gon, where n is the number of sides.

Sides Name

3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon

11 Undecagon

12 Dodecagon

Names of PolygonsNames of Polygons

Sides Name

3 Triangle

4 Quadrilateral*

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon

11 Undecagon

12 Dodecagon

Sides Name

13 Tridecagon

14 Tetradecagon

15 Pentadecagon

16 Hexadecagon

17 Heptadecaton

18 Octadecagon

19 Nonadecagon

20 Icosagon

100 Hectagon

1,000,000 Hecatommyriagon

*Also called a T

etragon

Example 6Example 6

Name each polygon.

A

B

D

C U

G

W

C

T

F

quadrilateral

hexagon

Example 7Example 7

When you buy a 42” television, how or where is that 42 inches measured?

42”

DiagonalDiagonal

DiagonalDiagonal

A diagonaldiagonal is a line segment that joins two nonconsecutive vertices of a polygon.

Example 8Example 8How many diagonals are there in an

octagon? (Do you really want to draw that? Heck no! In your notebook make a table and find a pattern!)

Convex & Concave PolygonsConvex & Concave Polygons

Convex & Concave PolygonsConvex & Concave Polygons

Convex polygonsConvex polygons have all their diagonals in the interior of the polygon.

Concave polygonsConcave polygons have at least one diagonal on the exterior of the polygon.

Example 9Example 9

Tell whether the figure is a polygon and whether it is convex or concave.

Equilateral PolygonEquilateral Polygon

An equilateral polygonequilateral polygon is a polygon in which all of its sides are congruent.

Equiangular PolygonEquiangular Polygon

An equiangular equiangular polygonpolygon is a polygon in which all its interior angles are congruent.

Regular PolygonRegular Polygon

A regular polygonregular polygon is a polygon that is equilateral and equiangular.

Example 10 Example 10

Classify the polygon by the number of sides. Tell whether the polygon is equilateral, equiangular, or regular. Explain your reasoning.

Example 11: SATExample 11: SAT

In the figure, RS = ST and the coordinates of S are (k, 3). What is the value of k?

y

x

(1, 0)

TS

R O-3

Example 12Example 12

Given that the figure is regular, find the values of x and y.

x=12, y=8