3.3: angles of polygons. what is a polygon? a polygon is a closed plane figure made up of three or...
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3.3: Angles of Polygons
What is a Polygon?
A polygon is a closed plane figure made up of three or more line segments that intersect only at their endpoints.
In a regular polygon all the sides are congruent, and all of the interior angles are congruent.
Interior angle measures of a polygon: The sum S of the interior angle measures of a polygon with n sides is
S (n 2) 1800
Concave and Convex Polygons
A polygon is convex when every line segment connecting any two vertices lies entirely inside the polygon.
A polygon is concave when at least one line segment connecting any two vertices lies outside the polygon.
Finding the Sum of Interior Angle Measures
Find the sum of the interior angle measures of the school crossing sign.
The sign is in the shape of a pentagon. It has 5 sides.
S (n 2) 1800
S (5 2) 1800
S 3 1800
S 5400
Practice
The honey comb has 6 sides
The web has 7 sides
Find the sum of the interior angle measures of the green polygon.
S (n 2) 1800
S (7 2) 1800
S 5 1800
S 9000
S (n 2) 1800
S (6 2) 1800
S 4 1800
S 7200
Finding an Interior Angle Measure of a Polygon
Step 1: The polygon has 7 sides. Find the sum of the interior angle measures.
Step 2: Write and solve an equation to find x.
Find the value of x.
S (7 2) 1800
S 5 1800
S 9000
140145115 120130128 x 900778 x 900
x 1220
Practice
Find the value of x.
S (6 2) 1800
S 4 1800
S 7200
S (4 2) 1800
S 2 1800
S 3600
S (5 2) 1800
S 3 1800
S 5400
135110 125120125 x 720615 x 720
x 1050
9080115 x 360285 x 360
x 750
145145110 2x 2x 540400 4x 5404x 140
x 350
Real Life Application
Step 1: A hexagon has 6 sides. Find the sum of the interior angle measures.
Step 2: Divide the sum by the number of interior angles, 6.
A cloud system discovered on Saturn is in the approximate shape of a regular hexagon. Find the measure of each interior angle of the hexagon.
S (n 2) 1800
S (6 2) 1800
S 4 1800
S 7200
7200 6 1200
Exterior Angle Measures of a Polygon
The sum of the measures of the exterior angles of a convex polygon is 3600.
Algebra w+x+y+z=3600
Finding Exterior Angle Measures
Write and solve an equation for z
Write and solve an equation for x
Find the measures of the exterior angles of each polygon.
x 5012791360x 268 360
x 920
124 z z 26 3602z 150 3602z 210
z 1050
The exterior angles measures are 1050 and (105+26)=1310
Practice
Find the measures of the interior angles of the polygon.
Write and solve an equation to find x.
909090 x x 3602702x 3602x 90
x 450
The exterior angle measures are 900, 450, 900, 900, and 450.