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Properties of PolygonsProperties of Polygons
The students will be able to describe the characteristics of a figure and to identify polygons.
Unit G 2
Polygon• A polygon is a closed plane figure
whose sides are segments that intersect only at their endpoints. No two sides with a common endpoint are collinear. • Which figure is a polygon?
A B C
B
Unit G 3
• The common endpoint of two sides is a vertex of the polygon.
• are sides of this polygon
Polygons• Each segment that forms the
polygon is a side of the polygon.
AB,BC,CD,DE,EA
A
E
D
C
B
A, B, C, D, E are vertices of this polygon.
Unit G 4
Parts of a Polygon• Two vertices of a polygon that are connected
by a side are called consecutive vertices. • Two sides of a polygon that are connected by
a vertex are called consecutive sides.
These are consecutive vertices.
These are consecutive sides
A line that connects two non-consecutive vertices is called a diagonal.
This is a diagonal
Unit G 5
Comparing Polygons• You can describe polygons by comparing
the lengths of the sides or the measures of the angles.
If all the sides of a polygon are con-gruent, then it is an equilateral polygon.
If all the angles of a polygon are con-gruent, then it is an equiangular polygon.
If a polygon is both equilateral and equi-angular then it is a regular polygon.
Unit G 6
Concave or Convex• A polygon can be either concave or
convex. • If you can draw diagonal so that it is
outside of the polygon, then the polygon is concave. If all diagonals are inside the polygon, then the polygon is convex.
• Are these polygons concave or convex?
concave
convex
Unit G 7
Classifying Polygons by the Number of Sides
A triangle is a polygon with 3 sides
A quadrilateral is a polygon with 4 sides
A pentagon is a polygon with 5 sides
Unit G 8
A hexagon is a polygon with 6 sides
A heptagon is a polygon with 7 sides
A octagon is a polygon with 8 sides
Classifying Polygons by the Number of Sides
Unit G 9
A polygon with 9 sides is a
A polygon with 10 sides is a
A polygon with 12 sides is a
A polygon with n sides is a
Classifying Polygons by the Number of Sides
Unit G 10FHS
Interior Angles of a Polygon
• What is the measure of the sum of all the angles of a triangle?
• If you can divide a polygon into triangles, then you can find out how many degrees are in the polygon.
Watch Out! - Every vertex of every triangle that you draw must also be a vertex of the polygon.
Unit G 11FHS
• Let’s try this method on this pentagon.
Interior Angles of a Polygon
First, draw the triangles, making sure we draw each line from the same vertex.Next, count the triangles and multiply by 180º
This is the measure of the sum of the angles of the polygon.
Unit G 12FHS
Polygon Interior Angle Sum Theorem
• We don’t always want to draw a picture of the polygon and divide it into triangles in order to find the sum of the angles in the polygon. So we have a formula we can use.
• The formula for the sum of the interior angle measures of a convex polygon with n sides is (n - 2)180º.
• The (n – 2) portion tells us how many triangles
can be drawn in that polygon.
Unit G 13FHS
Each Interior Angle• If the polygon is regular, then you can find
the measure of one interior angle by dividing the sum of the interior angles by the number of sides of the polygon.
• Example: Find one interior angle of a regular pentagon.
• First, you find the sum of the interior angles.
(n – 2)·180º = (5 – 2)·180º = 3·180º = 540º
Then divide by the number of angles
540º ÷ 5 = 108º
Unit G 14FHS
Polygon Exterior Angle Sum Theorem
• The sum of the exterior angle measures of any convex polygon is 360º.
• Example: Find the sum of the exterior angles of a convex octagon.
• How would we find one exterior angle of a regular octagon?
• Divide 360º by the number of sides of the octagon.
So 360º ÷ 8 =
Unit G 15FHS
How to name a Polygon• To name a polygon, list all of the
vertices in order in clockwise or counter-clockwise direction.
F
H
G
C
D
BA
E
Start anywhere and name all vertices in consecutive order:
CDEFGHAB or EDCBAHGF are two possible names for this octagon.
Unit G 16FHS
Inscribed Quadrilateral• Remember from Unit 1: The
measure of an inscribed angle is half the measure of its intercepted arc.
• If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.– ∠A and ∠C are
supplementary.E
A B
CD
– ∠B and ∠D are supplementary.