3.5 The Polygon Angle-Sum Theorems. Geometry Mr. Barnes . Objectives:. To Classify Polygons To find the sums of the measures of the interior and exterior angles of polygons. Definitions:. SIDE. Polygon —a plane figure that meets the following conditions:
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You can name a polygon by listing its vertices consecutively.
For instance, PQRST and QPTSR are two correct names for the polygon above.
If it is not, explain why.
Not D- because D has a side that isn’t a segment – it’s an arc.
Not E- because two of the sides intersect only one other side.
Not F- because some of its sides intersect more than two sides.Example 1: Identifying Polygons
Figures A, B, and C are polygons.
Convex if no line that contains a side of the polygon contains a point in the interior of the polygon.
Concave or non-convex if a line does contain a side of the polygon containing a point on the interior of the polygon.Convex or concave?
See how it doesn’t go on the
See how this crosses
a point on the inside?
A polygon is EQUILATERAL
If all of its sides are congruent.
A polygon is EQUIANGULAR
if all of its interior angles are congruent.
A polygon is REGULARif it is
equilateral and equiangular.
x°+ 2x° + 70° + 80° = 360° concave.
3x + 150 = 360
3x = 210
x = 70
Sum of the measures of int. s of
A quadrilateral is 360°
Combine like terms
Subtract 150 from each side.
Divide each side by 3.
80°Ex. : Interior Angles of a Quadrilateral
Find m Q and mR.
mQ = x° = 70°
mR = 2x°= 140°
►So, mQ = 70° and mR = 140°
Divide Each Polygon into triangles by drawing all diagonals that are possible from one vertex
Multiply the number of triangles by 180 to find the sum of the measures of the angles of each polygon.
Look for a pattern. Describe any that you have found.
Write a rule for the sum of the measures of the angles of an n-gonInvestigation Activity
Ex: Find the sum of the measures of the angles of a 15-gon
Sum = (n-2)180
= 2340Polygon Angle-Sum Theorem
The sum of the interior angles of a polygon is 9180. How many sides does the polygon have?
Sum = (n-2)180
9180 = (n-2)180
51 = n-2
53 = n
The polygon has 53 sides.Example
The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360.
An equilateral polygon has all sides congruent
An equiangular polygon has all angles congruent
A regular polygon is both equilateral and equiangular.Polygon Exterior Angle-Sum Theorem