3.6 Angles in Polygons Objectives: Warm-Up:

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3.6 Angles in Polygons Objectives: Warm-Up:
Develop and use formulas for the sums of the measures of interior and exterior angles of polygons Warm-Up: Hereβs a two part puzzle designed to prove that half of eleven is six. First rearrange two sticks to reveal the number eleven. Then remove half of the sticks to reveal the number six.

Convex Polygon: A polygon in which any line segment connecting two points of the polygon has no part outside the polygon.

Concave Polygon: A polygon that is not convex.

Consider the following Pentagon:
π π π π Divide the polygon into three triangular regions by drawing all the possible diagonals from one vertex. π π π π π Add the three expressions: Find each of the following: π<π+π<π+π<π= ______ π<π+π<π+π<π= ______ π<π+π<π+π<π= ______

Note: Polygon Triangle 3 1 180 Quadrilateral 4 2 360 Pentagon 5 3 540
You can form triangular regions by drawing all possible diagonals from a given vertex of any polygon # of triangular regions Sum of Interior angles Polygon # of sides Triangle 3 1 180 Quadrilateral 4 2 360 Pentagon 5 3 540 Hexagon 6 4 720 n-gon n nβ2 180(n-2)

The sum of the measures of the interior angles of a polygon with n sides is:

Note: Polygon Triangle 3 180 60 Quadrilateral 4 360 90 Pentagon 5 540
Recall that a regular polygon is on in which all the angles are congruent. Sum of Interior angles Measure of Interior angles Polygon # of sides Triangle 3 180 60 Quadrilateral 4 360 90 Pentagon 5 540 108 Hexagon 6 720 120 n-gon n 180(n-2) n 180(n-2)

The measure of an Interior Angle of a Regular Polygon with n sides is:

Exterior Angle Sums in Polygons

Polygon Triangle 3 540 180 360 Quadrilateral 4 720 360 360 Pentagon 5
Sum of interior & exterior angles Sum of Interior angles Sum of Exterior angles Polygon # of sides Triangle 3 540 180 360 Quadrilateral 4 720 360 360 Pentagon 5 900 540 360 Hexagon 6 1080 720 360 n-gon n 180n 360 180(n-2)

Theorem Sum of the measures of the Exterior Angles of a Polygon is: πππ π

For a Convex Polygon For a Regular Polygon Polygon Number of Sides Number of Ξ Regions Sum of Interior Angles Sum of Exterior Angles Sum of Int & Ext Angles Measure of Interior Angles Measure Exterior Angles Triangle 3 1 180 360 540 60 120 Quadrilateral 4 2 720 90 Pentagon 5 900 108 72 Hexagon 6 1080 Heptagon 7 1260 128.6 51.4 Octagon 8 1440 135 45 Nonagon 9 1620 140 40 Decagon 10 1800 144 36 11-gon 11 1980 147.3 32.7 Dodecagon 12 2160 150 30 13-gon 13 2340 152.3 27.2 n-gon n n-2 180(n-2) 180n πππ(π§βπ) π§ πππβ πππ(π§βπ) π§

Find the indicated angle measures.
πππ π πππ π ππ π π π π π π π ππ π πππ π

Find the indicated angle measures.
π π ππ π ππ π π π ππ π πππ π πππ π

Find the indicated angle measures.
πππ π π π πππ π ππ π πππ π

Find the indicated angle measures.
π π πππ π πππ π ππ π πππ π

A rectangle An equilateral triangle A regular dodecagon An equiangular
For each polygon determine the measure of an interior angle and the measure of an exterior angle. A rectangle An equilateral triangle A regular dodecagon An equiangular pentagon

An interior angle measure of a regular polygon is given
An interior angle measure of a regular polygon is given. Find the number of sides of the polygon πππ π πππ π πππ π

An exterior angle measure of a regular polygon is given
An exterior angle measure of a regular polygon is given. Find the number of sides of the polygon ππ π ππ π ππ π

Find the indicated angle measure.
(ππ) π (ππ) π (π) π (ππ) π

Find the indicated angle measure.
(π π + ππ+ππ) π (ππ+ππ) π (ππβππ) π (π π +ππ) π

Find the indicated angle measure.
( ππβππ) π ( ππ+ππ) π ( ππβππ) π ( ππ+π) π ( ππ+ππ) π