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Quadrilaterals, Diagonals, and Angles of Polygons
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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
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Classifying Polygons
Number of Sides
Name of Polygon
Number of Sides
Name of Polygon
3 Triangle 4 Quadrilateral
5 Pentagon 6 Hexagon
7 Heptagon 8 Octagon
9 Nonagon 10 Decagon
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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?
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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!
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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
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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 Sketch of Figure
Number of Triangles
Sum of Angle Measurements
3 1 1(180) = 180
4 2 2(180) = 360
5
6
7
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Angles of Polygons Mini-Lab
• Draw a pentagon with diagonals from one vertex to each opposing vertex
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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 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
7
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Angles of Polygons Mini-Lab
• Draw a hexagon with diagonals from one vertex to each opposing vertex
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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 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
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Angles of Polygons Mini-Lab
• Draw a heptagon with diagonals from one vertex to each opposing vertex
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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
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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!
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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
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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
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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
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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
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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
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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 = 23402340 / 15 = 156
21 x 180 = 3780
3780 / 23 = 164.35
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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 = 23402340 / 15 = 156
21 x 180 = 37803780 / 23 = 164.35
28 x 180 = 5040
5040 / 30 = 168
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Homework
• Skill 4: Polygons (both sides)
• 6-3 Skills Practice: Polygons and Angles
• DUE TOMORROW!