Polygon Angle Sum Theorems Geometry 3 4 Vocabulary

Polygon Angle Sum Theorems Geometry 3 -4

Vocabulary

• Name a polygon with consecutive vertices Polygons

• Diagonal – A line segment that connects two non-consecutive vertices • Convex – No diagonal goes outside the polygon • Concave – At least one diagonal is outside the polygon Vocabulary

• Equilateral – A polygon with all sides congruent • Equiangular – A polygon with all angles congruent • Regular Polygon – A polygon that is both equilateral and equiangular Vocabulary

Equilateral Triangle 60° Angle Exploration

Square 90° Angle Exploration

Regular Pentagon 108° Angle Exploration

Regular Hexagon 120° Angle Exploration

Regular Heptagon 129° Angle Exploration

Regular Octagon 135° Angle Exploration

Regular Nonagon 140° Angle Exploration

Regular Decagon 144° Angle Exploration

Regular Dodecagon 147° Angle Exploration

• Summarize our results Is there a correlation between the number of sides and the degrees? Angle Exploration Sides 3 4 5 6 7 8 9 10 11 Degrees 60° 90° 108° 120° 129° 135° 140° 144° 147°

• What is the sum of the angles in each of the polygons Angle Exploration

• Summarize our results Total Degrees in each polygon. Do you see a pattern? Angle Exploration Sides 3 4 5 6 7 8 9 10 11 Total 180° 360° 540° 720° 900° 1080° 1260° 1440° 1620° Degrees 60° 90° 108° 120° 129° 135° 140° 144° 147°

• The sum of the angles of the interior angles of a convex ngon is (n-2) • 180° Polygon Angle Sum Theorem

• What are the angle measures if the polygon is regular • (n-2) • 180°/n Interior Angles in a Polygon

X=85

X=97

8

36

71 19. 5

60

17

104

X=96

X=117

Polygon Exterior Angle Sum Geometry 3 -4 b

2 1 Vertical Pair Review – Name the angle relationship

160° 20° Supplementary Angles Review – Name the angle relationship

2 1 Linear Pair Review – Name the angle relationship

35° 55° Complementary Angles Review – Name the angle relationship

2 1 Vertical Pair Review – Name the angle relationship

60° 30° Complementary Angles Review – Name the angle relationship

1 2 Linear Pair Review – Name the angle relationship

90° Supplementary Angles Review – Name the angle relationship

Any of the six angles inside the polygon What is an interior angle?

When the sides of the polygon are extended, the angle that is adjacent to the interior angle What is an exterior angle?

Spot the exterior angle

Which side do you extend to create an exterior angle? Exterior Angles

Either one, these two angles are congruent, because they are vertical angles. Exterior Angles

What is the value of all the exterior angles in any polygon? Exterior angle

• Get your supplies – Printer Paper – Straightedge – Protractor Exterior Angle Investigation

• Draw a large polygon on your paper Pentagon or larger • Extend each of its sides to form exterior angles Exterior Angle Investigation

• Measure all of the interior angles except one. Write the value of the interior angle on the angle. • Use the polygon sum conjecture to calculate the measure of the last angle before you measure it. Exterior Angle Investigation

• Use the linear pair conjecture to calculate the value of each exterior angle. • Write down the value on the angle • Add up all the exterior measures Exterior Angle Investigation

• The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360° Polygon Exterior Angle Sum Theorem

What is the measure of an exterior angle in a regular polygon? Exterior angle

• The measure of each exterior angle of a regular n-gon is 360°/n Exterior Angle Theorem

• You can find the measure of each interior angle of an equiangular n-gon by using either of these formulas: 180° - 360°/n (180°(n-2))/n Equiangular Polygon Conjecture

• Pages 147 – 150 • 1 – 4, 12, 13, 17, 20, 47 – 49, 54, 71, 73 - 79 Homework