Section 3-4 Polygon Angle-Sum Theorem SPI 32A: Identify properties of plane figures from information given in a diagram

Objectives:• Classify Polygons• Find the sums of the measures of the interior and exterior angle of polygons

Polygon: • closed plane figure with at least 3 sides that are segments• the sides intersect only at their endpoints• no adjacent sides are collinear

Classify Polygons

Name Polygons By Their:

VerticesStart at any vertex and list the vertices consecutively in a clockwise direction (ABCDE or CDEAB, etc)

SidesName by line segment naming convention

AnglesName by angle naming convention

, , , ,A B C D E

, , , ,AB BC CD DE EA

Classify Polygons by the Number of Sides

Number of Sides Name

3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

8 Octagon

9 Nonagon

10 Decagon

12 Dodecagon

n n-gon

Classify Polygons as Convex or Concave

Convex PolygonHas no diagonals with points outside the polygon

Concave PolygonHas at least one diagonal outside the polygon

Starting with any side, count the number of sides clockwise around the figure. Because the polygon has 12 sides, it is a dodecagon.

Classify the polygon below by its sides. Identify it as convex

or concave.

Think of the polygon as a star. If you draw a diagonal connecting two points of the star that are next to each other,that diagonal lies outside the polygon, so the dodecagon is concave.

Classify Polygons as Convex or Concave

Triangle Angle-Sum Theorem

1. Draw and cut out a triangle.

2. Number the angles and tear them off.

3. Place the angles adjacent to each other.

4. Compare your results with others. What do you observe about the sum of the angles of a triangle?

Triangle Angle-Sum TheoremThe sum of the measures of the angles of a triangle measure 180º.

Polygon Angle-Sum Theorem

Use the Triangle Angle-Sum Theorem to find the sum of the measures of the angles of a polygon.

1. Sketch convex polygons with 4, 5, 6, 7, and 8 sides. Construct a table to record your data in order to look for a pattern or rule to find the sum of the measures of the angles of an n-gon.

2. Divide each polygon into triangles by drawing all diagonals that are possible from one vertex.

3. Multiply the number of triangles by 180 to find the sum of the measures of the angles of each polygon.

Polygon Number sides

(n)

# of Triangles Sum of Interior angle measures (___∙ 180= ___)

4 2 2 ∙ 180 = 360

n (n - 2) ∙ 180

Polygon Angle-Sum Theorem

Theorem 3-9: Polygon Angle-Sum Theorem

The sum of the measures of the angles of an n-gon is (n - 2) 180.

A decagon has 10 sides, so n = 10.

Sum = (n – 2)(180) Polygon Angle-Sum Theorem

= (10 – 2)(180) Substitute 10 for n.

= 8 • 180 Simplify.

= 1440

Find the sum of the measures of the angles of a decagon.

Polygon Angle-Sum Theorem

The sum of the measures of the angles of a given polygon is 720. How can you use the Polygon Angle-Sum Theorem to find the number of sides in the polygon?

Sum = (n – 2) 180 Write the Equation

720 = (n – 2) 180 Sub. In known values

720 = 180n – 360 Simplify

1080 = 180n Addition Prop of EQ

6 = n Hexagon (6 sides)

m X + m Y + m Z + m W = (4 – 2)(180) Polygon Angle-Sum Theorem

m X + m Y + 90 + 100 = 360 Substitute.

m X + m Y + 190 = 360 Simplify.

m X + m Y = 170 Subtract 190 from each side.

2m X = 170 Simplify.

m X = 85 Divide each side by 2.

m X + m X = 170 Substitute m X for m Y.

The figure has 4 sides, so n = 4.

Find m X in quadrilateral XYZW.

Use the Polygon Angle-Sum Theorem

Polygon Exterior Angle-Sum Theorem

Equilateral Polygon: • all sides are congruent

Equiangular Polygon: • all angles are congruent

Regular Polygon: • is both equilateral and equiangular

Because supplements of congruent angles are congruent, all the angles marked 1 have equal measures.

The hexagon is regular, so all its angles are congruent.

An exterior angle is the supplement of a polygon’s angle because they are adjacent angles that form a straight angle.

Below is a regular hexagon game board packaged in a

rectangular box. Explain how you know that all the

angles labeled 1 have equal measures.

Real-world Connection