Download The Polygon Angle-Sum Theorems

The Polygon Angle-Sum Theorems
Lesson 3-5
Additional Examples
Geometry
Name the polygon. Then identify its vertices, sides,
and angles.
The polygon can be named clockwise or counterclockwise,
starting at any vertex.
Possible names are ABCDE and EDCBA.
Its vertices are A, B, C, D, and E.
Its sides are AB or BA, BC or CB, CD or DC, DE or ED, and EA or AE.
Its angles are named by the vertices, A (or EAB or BAE),
B (or ABC or CBA), C (or BCD or DCB),
D (or CDE or EDC), and E (or DEA or AED).
The Polygon Angle-Sum Theorems
Lesson 3-5
Additional Examples
Geometry
Classify the polygon below by its sides. Identify it as convex
or concave.
Starting with any side, count the number of sides clockwise
around the figure. Because the polygon has 12 sides,
it is a dodecagon.
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.
The Polygon Angle-Sum Theorems
Lesson 3-5
Geometry
Additional Examples
Find the sum of the measures of the angles of a decagon.
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
The Polygon Angle-Sum Theorems
Lesson 3-5
Geometry
Additional Examples
Find m
X in quadrilateral XYZW.
The figure has 4 sides, so n = 4.
m
X + m Y + m Z + m W = (4 – 2)(180)
m X + m Y + 90 + 100 = 360
m
X+m
m
m
Y + 190 = 360
X+m
X+m
2m
m
Polygon Angle-Sum Theorem
Substitute.
Simplify.
Y = 170
X = 170
X = 170
Subtract 190 from each side.
Substitute m X for m Y.
Simplify.
X = 85
Divide each side by 2.
The Polygon Angle-Sum Theorems
Lesson 3-5
Additional Examples
Geometry
A regular hexagon is inscribed in a rectangle. Explain how you
know that all the angles labeled 1 have equal measures.
Sample: 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.
Because supplements of congruent angles are congruent, all the angles
marked 1 have equal measures.