Transcript Slide 1

Geometry
Chapter 6
Quadrilaterals
Lesson 1:
Angles of Polygons
Warm Up
1. A
?
2. A
?
is a three-sided polygon. triangle
is a four-sided polygon. quadrilateral
Evaluate each expression for n = 6.
3. (n – 4) 12
24
4. (n – 3) 90
270
Solve for a.
5. 12a + 4a + 9a = 100 4
Your Math Goal Today…
Classify polygons based on their sides
and angles.
Find and use the measures of interior
and exterior angles of polygons.
Vocabulary
side of a polygon
vertex of a polygon
diagonal
regular polygon
concave
convex
Each segment that forms a polygon is a side of
the polygon. The common endpoint of two sides
is a vertex of the polygon. A segment that
connects any two nonconsecutive vertices is a
diagonal.
You can name a polygon
by the number of its
sides. The table shows
the names of some
common polygons.
Remember!
A polygon is a closed plane figure formed by
three or more segments that intersect only at
their endpoints.
Example 1A: Identifying Polygons
If it is a polygon, name it by the number of
sides.
polygon, hexagon
Example 1B: Identifying Polygons
If it is a polygon, name it by the number of
sides.
polygon, heptagon
Example 1C: Identifying Polygons
If it is a polygon, name it by the number of
sides.
not a polygon
In Your Notes
If it is a polygon, name it by the number of its
sides.
not a polygon
In Your Notes
If it is a polygon, name it by the number of its
sides.
polygon, nonagon
In Your Notes
If it is a polygon, name it by the number of its
sides.
not a polygon
All the sides are congruent in an equilateral polygon.
All the angles are congruent in an equiangular
polygon. A regular polygon is one that is both
equilateral and equiangular. If a polygon is not
regular, it is called irregular.
A polygon is concave if any part of a diagonal
contains points in the exterior of the polygon. If no
diagonal contains points in the exterior, then the
polygon is convex. A regular polygon is always
convex.
Example 2A: Classifying Polygons
Tell whether the polygon is regular or
irregular. Tell whether it is concave or convex.
irregular, convex
Example 2B: Classifying Polygons
Tell whether the polygon is regular or
irregular. Tell whether it is concave or convex.
irregular, concave
Example 2C: Classifying Polygons
Tell whether the polygon is regular or
irregular. Tell whether it is concave or convex.
regular, convex
In Your Notes
Tell whether the polygon is regular or irregular.
Tell whether it is concave or convex.
regular, convex
In Your Notes
Tell whether the polygon is regular or irregular.
Tell whether it is concave or convex.
irregular, concave
To find the sum of the interior angle measures of a
convex polygon, draw all possible diagonals from
one vertex of the polygon. This creates a set of
triangles. The sum of the angle measures of all the
triangles equals the sum of the angle measures of
the polygon.
Remember!
By the Triangle Sum Theorem, the sum of the
interior angle measures of a triangle is 180°.
In each convex polygon, the number of triangles
formed is two less than the number of sides n. So
the sum of the angle measures of all these triangles
is (n — 2)180°.
Example 3A: Finding Interior Angle Measures and
Sums in Polygons
Find the sum of the interior angle measures of a
convex heptagon.
(n – 2)180°
Polygon  Sum Thm.
(7 – 2)180°
A heptagon has 7 sides,
so substitute 7 for n.
900°
Simplify.
Example 3B: Finding Interior Angle Measures and
Sums in Polygons
Find the measure of each interior angle of a
regular 16-gon.
Step 1 Find the sum of the interior angle measures.
(n – 2)180°
Polygon  Sum Thm.
(16 – 2)180° = 2520°
Substitute 16 for n
and simplify.
Step 2 Find the measure of one interior angle.
The int. s are , so divide by 16.
Example 3C: Finding Interior Angle Measures and
Sums in Polygons
Find the measure of each
interior angle of pentagon
ABCDE.
(5 – 2)180° = 540° Polygon  Sum Thm.
Polygon 
mA + mB + mC + mD + mE = 540° Sum Thm.
35c + 18c + 32c + 32c + 18c = 540
135c = 540
c=4
Substitute.
Combine like terms.
Divide both sides by 135.
Example 3C Continued
mA = 35(4°) = 140°
mB = mE = 18(4°) = 72°
mC = mD = 32(4°) = 128°
In Your Notes
Find the sum of the interior angle measures of
a convex 15-gon.
(n – 2)180°
Polygon  Sum Thm.
(15 – 2)180° A 15-gon has 15 sides, so
substitute 15 for n.
2340°
Simplify.
In Your Notes
Find the measure of each interior angle of a
regular decagon.
Step 1 Find the sum of the interior angle measures.
(n – 2)180°
Polygon  Sum Thm.
(10 – 2)180° = 1440°
Substitute 10 for n
and simplify.
Step 2 Find the measure of one interior angle.
The int. s are , so divide by 10.
In the polygons below, an exterior angle has been
measured at each vertex. Notice that in each case,
the sum of the exterior angle measures is 360°.
Remember!
An exterior angle is formed by one side of a
polygon and the extension of a consecutive side.
Example 4A: Finding Interior Angle Measures and
Sums in Polygons
Find the measure of each exterior angle of a
regular 20-gon.
A 20-gon has 20 sides and 20 vertices.
sum of ext. s = 360°.
measure of one ext.  =
Polygon  Sum Thm.
A regular 20-gon
has 20  ext. s, so
divide the sum by
20.
The measure of each exterior angle of a regular
20-gon is 18°.
Example 4B: Finding Interior Angle Measures and
Sums in Polygons
Find the value of b in polygon
FGHJKL.
Polygon Ext.  Sum Thm.
15b° + 18b° + 33b° + 16b° + 10b° + 28b° = 360°
120b = 360
b=3
Combine like terms.
Divide both sides by 120.
In Your Notes
Find the measure of each exterior angle of a
regular dodecagon.
A dodecagon has 12 sides and 12 vertices.
sum of ext. s = 360°.
measure of one ext.
Polygon  Sum Thm.
A regular dodecagon
has 12  ext. s, so
divide the sum by
12.
The measure of each exterior angle of a regular
dodecagon is 30°.
In Your Notes
Find the value of r in polygon JKLM.
4r° + 7r° + 5r° + 8r° = 360° Polygon Ext.  Sum Thm.
24r = 360
r = 15
Combine like terms.
Divide both sides by 24.
Example 5: Art Application
Ann is making paper stars for
party decorations. What is the
measure of 1?
1 is an exterior angle of a regular
pentagon. By the Polygon Exterior
Angle Sum Theorem, the sum of the
exterior angles measures is 360°.
A regular pentagon has 5 
ext. , so divide the sum by 5.
In Your Notes
What if…? Suppose the
shutter were formed by 8
blades instead of 10
blades. What would the
measure of each exterior
angle be?
CBD is an exterior angle of a regular octagon. By
the Polygon Exterior Angle Sum Theorem, the sum
of the exterior angles measures is 360°.
A regular octagon has 8  ext.
, so divide the sum by 8.