Geometry 3-4 Polygon Angle Sum Theorems

Vocabulary

Polygons Name a polygon with consecutive vertices

Vocabulary 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

Angle Exploration 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 SidesDegrees 360° 490° 5108° 6120° 7129° 8135° 9140° 10144° 11147° Is there a correlation between the number of sides and the degrees?

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

Angle Exploration Summarize our results SidesTotalDegrees 3180°60° 4360°90° 5540°108° 6720°120° 7900°129° 81080°135° 91260°140° °144° °147° Total Degrees in each polygon. Do you see a pattern?

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

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

X=85

X=97

8

36

19.571

60

17

104

X=96

X=117

Geometry 3-4b Polygon Exterior Angle Sum

Review – Name the angle relationship 1 2 Vertical Pair

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

Review – Name the angle relationship 1 2 Linear Pair

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

Review – Name the angle relationship 1 2 Vertical Pair

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

Review – Name the angle relationship 1 2 Linear Pair

Review – Name the angle relationship 90° Supplementary Angles 90°

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

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

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

Spot the exterior angle

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

Exterior Angle Investigation 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

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

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

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

Equiangular Polygon Conjecture 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

Homework Pages 147 – – 4, 12, 13, 17, 20, 47 – 49, 54, 71,

Homework Pages 147 – , 13, 17, 20, 54, 71,