Download Quadrilaterals, Diagonals, and Polygons

Quadrilaterals, Diagonals, and
Angles of Polygons
Quadrilaterals, Diagonals, and
Angles of Polygons
• A Polygon is a simple closed plane figure,
having three or more line segments as
sides
• A Quadrilateral is any four-sided closed
plane figure
• A Diagonal a line segment that connects
one vertex to another (but not next to it) on
a polygon
Classifying Polygons
Number of
Sides
3
Name of
Polygon
Triangle
Number of
Name of
Sides
Polygon
4
Quadrilateral
5
Pentagon
6
Hexagon
7
Heptagon
8
Octagon
9
Nonagon
10
Decagon
Quadrilateral Angles
• We know that the interior angles of a
triangle add up to 180 degrees
• How many degrees are in the interior
angles of a quadrilateral?
Quadrilateral Angles
• If we draw a diagonal from one vertex across to
the opposite vertex, we see that we have formed
two triangles
• Therefore, the sum of two triangles will give you
the measure of the interior angles of a
quadrilateral
• 180 + 180 = 360 degrees!
Quadrilateral Angles Checkpoint
• Find the missing angle of a quadrilateral
with the following measures:
m 1 = 117
m 2 = 110
m 3 = 75
m 4 = 117 + 110 + 75 + x = 360
302 + x = 360
x = 58
Angles of Polygons Mini-Lab
• Let’s explore this knowledge and how it relates
to the angles of other polygons
• Copy and complete the table below:
Number of Sides
Number of
Triangles
Sum of Angle
Measurements
3
1
1(180) = 180
4
2
2(180) = 360
5
6
7
Sketch of
Figure
Angles of Polygons Mini-Lab
• Draw a pentagon with diagonals from one
vertex to each opposing vertex
Angles of Polygons Mini-Lab
• Let’s explore this knowledge and how it relates
to the angles of other polygons
• Copy and complete the table below:
Number of Sides
Number of
Triangles
Sum of Angle
Measurements
3
1
1(180) = 180
4
2
2(180) = 360
5
3
3(180) = 540
6
7
Sketch of
Figure
Angles of Polygons Mini-Lab
• Draw a hexagon with diagonals from one
vertex to each opposing vertex
Angles of Polygons Mini-Lab
• Let’s explore this knowledge and how it relates
to the angles of other polygons
• Copy and complete the table below:
Number of Sides
Number of
Triangles
Sum of Angle
Measurements
3
1
1(180) = 180
4
2
2(180) = 360
5
3
3(180) = 540
6
4
4(180) = 720
7
Sketch of
Figure
Angles of Polygons Mini-Lab
• Draw a heptagon with diagonals from one
vertex to each opposing vertex
Angles of Polygons Mini-Lab
• Let’s explore this knowledge in how it relates to
the angles of other polygons
• Copy and complete the table below:
Number of Sides
Sketch of
Figure
Number of
Triangles
Sum of Angle
Measurements
3
1
1(180) = 180
4
2
2(180) = 360
5
3
3(180) = 540
6
4
4(180) = 720
7
5
5(180) = 900
Angles of Polygons Mini-Lab
• What patterns do you see as a result of
our experiment?
• The number of triangles in any polygon is
always two less than the number of sides.
• Therefore, if n = the number of sides of the
polygon; the sum of interior angles of any
polygon can be expressed as (n – 2)180!
Angles of Polygons Checkpoint
• Find the sum of the measures of the
interior angles of each polygon:
15-gon?
23-gon?
30-gon?
(15-sided figure)
(23-sided figure)
(30-sided figure)
13 x 180 = 2340
Angles of Polygons Checkpoint
• Find the sum of the measures of the
interior angles of each polygon:
15-gon?
23-gon?
30-gon?
(15-sided figure)
(23-sided figure)
(30-sided figure)
13 x 180 = 2340
21 x 180 = 3780
Angles of Polygons Checkpoint
• Find the sum of the measures of the
interior angles of each polygon:
15-gon?
23-gon?
30-gon?
(15-sided figure)
(23-sided figure)
(30-sided figure)
13 x 180 = 2340
21 x 180 = 3780
28 x 180 = 5040
Regular Polygons
• A regular polygon is one that is equilateral
(all sides congruent) and equiangular (all
angles congruent)
• Polygons that are not regular are said to
be irregular
• If the formula for finding the sum of
measures of interior angles of a polygon is
(n-2)180, how would you find the measure
of each angle of a regular polygon?
( n – 2 )180
n
Regular Polygons Checkpoint
• Find the sum of the measures of the
interior angles of each regular polygon and
the measure of each individual angle:
15-gon?
23-gon?
30-gon?
(15-sided figure)
(23-sided figure)
(30-sided figure)
13 x 180 = 2340
2340 / 15 = 156
Regular Polygons Checkpoint
• Find the sum of the measures of the
interior angles of each regular polygon and
the measure of each individual angle:
15-gon?
23-gon?
30-gon?
(15-sided figure)
(23-sided figure)
(30-sided figure)
13 x 180 = 2340
2340 / 15 = 156
21 x 180 = 3780
3780 / 23 = 164.35
Regular Polygons Checkpoint
• Find the sum of the measures of the
interior angles of each regular polygon and
the measure of each individual angle:
15-gon?
23-gon?
30-gon?
(15-sided figure)
(23-sided figure)
(30-sided figure)
13 x 180 = 2340
2340 / 15 = 156
21 x 180 = 3780
3780 / 23 = 164.35
28 x 180 = 5040
5040 / 30 = 168
Homework
• Skill 4: Polygons (both sides)
• 6-3 Skills Practice: Polygons and Angles
• DUE TOMORROW!