DrillDrill1) If two angles of a triangle have a

sum of 85 degrees find the third angle.

2) The three angles of a triangle are 2x, 3x, and 2x + 40 find each angle.

2.2 PolygonsPolygons

Polygon Is a closed figure with at least three

sides, so that each segment intersects exactly two segments at their endpoints.

Polygon TerminologyPolygon Terminology

Sides Vertex

Interior

A

B

C D

E

FDiagonal

ConsecutiveVertices

A polygon can also be classified as convex or concave.

If all of the diagonalslie in the interior of the figure, then the

polygon is ______.convex

If any part of a diagonal liesoutside of the figure, then thepolygon is _______.concave

Naming PolygonsNaming Polygons

Types of Polygons# of Sides Name/Draw

3

4

5

6

7

8

9

10

12

QUADRILATERAL

PENTAGON

HEXAGON

OCTAGON

HEPTAGON

NONAGON

DECAGON

DODECAGON

TRIANGLE

Diagonals and Angle Measure Diagonals and Angle Measure

Convex

Polygon

Number

of Sides

Number of Diagonals from One Vertex

Number of

Triangles

Sum of

Interior Angles

quadrilateral 4 1 2 2(180) = 360

1) Draw a convex quadrilateral.

2) Choose one vertex and draw all possible diagonals from that vertex.

3) How many triangles are formed?

Make a table like the one below.

Diagonals and Angle Measure Diagonals and Angle Measure

Convex

Polygon

Number

of Sides

Number of Diagonals from One Vertex

Number of

Triangles

Sum of

Interior Angles

quadrilateral 4 1 2 2(180) = 360

1) Draw a convex pentagon.

2) Choose one vertex and draw all possible diagonals from that vertex.

3) How many triangles are formed?

pentagon 5 2 3 3(180) = 540

Diagonals and Angle Measure Diagonals and Angle Measure

Convex

Polygon

Number

of Sides

Number of Diagonals from One Vertex

Number of

Triangles

Sum of

Interior Angles

quadrilateral 4 1 2 2(180) = 360

1) Draw a convex hexagon.

2) Choose one vertex and draw all possible diagonals from that vertex.

3) How many triangles are formed?

pentagon 5 2 3 3(180) = 540

hexagon 6 3 4 4(180) = 720

Diagonals and Angle Measure Diagonals and Angle Measure

Convex

Polygon

Number

of Sides

Number of Diagonals from One Vertex

Number of

Triangles

Sum of

Interior Angles

quadrilateral 4 1 2 2(180) = 360

1) Draw a convex heptagon.

2) Choose one vertex and draw all possible diagonals from that vertex.

3) How many triangles are formed?

pentagon 5 2 3 3(180) = 540

hexagon 6 3 4 4(180) = 720

heptagon 7 4 5 5(180) = 900

Diagonals and Angle Measure Diagonals and Angle Measure

Convex

Polygon

Number

of Sides

Number of Diagonals from One Vertex

Number of

Triangles

Sum of

Interior Angles

quadrilateral 4 1 2 2(180) = 360

1) Any convex polygon.

2) All possible diagonals from one vertex.

3) How many triangles?

pentagon 5 2 3 3(180) = 540

hexagon 6 3 4 4(180) = 720

heptagon 7 4 5 5(180) = 900

n-gon n n - 3 n - 2 (n – 2)180

Theorem 10-1If a convex polygon has n sides, then the sum of the measure of its interior angles is (n – 2)180.

Diagonals and Angle Measure Diagonals and Angle Measure

57°48°

74°

55°54°

72°

In §7.2 we identified exterior angles of triangles.

Likewise, you can extend the sides of anyconvex polygon to form exterior angles.

The figure suggests a method for finding thesum of the measures of the exterior angles of a convex polygon.

When you extend n sides of a polygon,

n linear pairs of angles are formed.

The sum of the angle measures in each linear pair is 180.

sum of measure of exterior angles

sum of measures of linear pairs

sum of measures of interior angles=

=

–

–n•180 180(n – 2)= –180n 180n + 360

= 360sum of measure of exterior angles

Polygon Interior Angle-Sum TheoremPolygon Interior Angle-Sum Theorem

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

Polygon Exterior Angle-Sum TheoremPolygon Exterior Angle-Sum Theorem

The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360.

HomeworkHomework

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