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3.5 The Polygon Angle-Sum Theorems

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3.5 The Polygon Angle-Sum Theorems

Geometry

Mr. Barnes

- To Classify Polygons
- To find the sums of the measures of the interior and exterior angles of polygons.

SIDE

- Polygon—a plane figure that meets the following conditions:
- It is formed by 3 or more segments called sides, such that no two sides with a common endpoint are collinear.
- Each side intersects exactly two other sides, one at each endpoint.

- Vertex – each endpoint of a side. Plural is vertices.
You can name a polygon by listing its vertices consecutively.

For instance, PQRST and QPTSR are two correct names for the polygon above.

State whether the figure is a polygon.

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.

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.

See how it doesn’t go on the

Inside-- convex

See how this crosses

a point on the inside?

Concave.

Identify the polygon and state whether it is convex or concave.

CONCAVE

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.

CONVEX

x°+ 2x° + 70° + 80° = 360°

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°

70°

2x°

x°

Find m Q and mR.

mQ = x° = 70°

mR = 2x°= 140°

►So, mQ = 70° and mR = 140°

Sketch polygons with 4, 5, 6, 7, and 8 sides

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-gon

The sum of the measures of the angles of an n-gon is

(n-2)180

Ex: Find the sum of the measures of the angles of a 15-gon

Sum = (n-2)180

= (15-2)180

= (13)180

= 2340

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.

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.