1.5: describe angle pair relationships 1.6: classify polygons objectives: 1.to use special angle...
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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
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