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Lesson 6-1
Angles of Polygons
5-Minute Check on Chapter 5
1. State whether this sentence is always, sometimes, or never true.
The three altitudes of a triangle intersect at a point inside the
triangle. Sometimes – when all angles are acute
2. Find n and list the sides of ΔPQR in order from shortest to longest if
mP = 12n – 15, mQ = 7n + 26, and mR = 8n – 47.
all angle add to 180 so n = 8; PQ < QR < PR
3. State the assumption you would make to start an indirect proof of
the statement.
If –2x ≥ 18, then x ≤ –9.
x > -9
4. Find the range for the measure of the third side of a triangle given
that the measures of two sides are 43 and 29.
43 – 29 = 14 < n < 72 = 29 + 43
5. Write an inequality relating mABD and mCBD.
Since 11 > 10, then mABD < mCBD
6. Write an equation that you can use to find
the measures of the angles of the triangle.
111 = 3x + (x – 5)
Click the mouse button or press the
Space Bar to display the answers.
Objectives
• Find and use the sum of the measures of the
interior angles of a polygon
– Sum of Interior angles = (n-2) • 180
– One Interior angle = (n-2) • 180 / n
• Find and use the sum of the measures of the
exterior angles of a polygon
– Sum of Exterior angles = 360
– One Exterior angle = 360/n
– Exterior angle + Interior angle = 180
Vocabulary
• Diagonal – a segment that connects any two
nonconsecutive vertices in a polygon.
Angles in a Polygon
3
2
4
1
5
Octagon
n=8
6
8
7
8 triangles @ 180° - 360° (center angles)
= (8-2) • 180 = 1080
Sum of Interior angles = (n-2) • 180
Angles in a Polygon
Sum of Interior Angles:
Octagon
n=8
(n – 2) * 180 where n is number of sides
so each interior angle is
(n – 2) * 180
n
Interior Angle
Sum of Exterior Angles:
360
so each exterior angle is
360
n
Exterior Angle
Octagon
Sum of Exterior Angles:
Sum of Interior Angles:
One Interior Angle:
One Exterior Angle:
360
1080
135
45
Interior Angle + Exterior Angle = 180
Polygons
Sides
Name
Sum of
Interior
Angles
One
Interior
Angle
3
Triangle
180
60
360
120
4
Quadrilateral
360
90
360
90
5
Pentagon
540
108
360
72
6
Hexagon
720
120
360
60
7
Heptagon
900
129
360
51
8
Octagon
1080
135
360
45
9
Nonagon
1260
140
360
40
10
Decagon
1440
144
360
36
12
Dodecagon
1800
150
360
30
n
N - gon
360
360 ∕ n =
(n-2) * 180 180 – Ext 
Sum Of
One
Exterior Exterior
Angles Angles
Angle Theorems
ARCHITECTURE A mall is
designed so that five walkways
meet at a food court that is in the
shape of a regular pentagon. Find
the sum of measures of the
interior angles of the pentagon.
Since a pentagon is a convex polygon, we can use the
Angle Sum Theorem.
Interior Angle Sum Theorem
Simplify.
Answer: The sum of the measures of the angles is 540.
The measure of an interior angle of a regular polygon
is 135. Find the number of sides in the polygon.
Use the Interior Angle Sum Theorem to write an equation
to solve for n, the number of sides.
Interior Angle Sum Theorem
Distributive Property
Subtract 135n from each side.
Add 360 to each side.
Divide each side by 45.
Answer: The polygon has 8 sides.
SHORT CUT!!
The measure of an interior angle of a regular polygon
is 135. Find the number of sides in the polygon.
Exterior angle = 180 – Interior angle = 45
360
360
n = --------- = ------- = 8
Ext 
45
The measure of an interior angle of a regular polygon
is 144. Find the number of sides in the polygon.
Answer: The polygon has 10 sides.
Find the measure of each interior angle.
Since n = 4 the sum of the measures of the interior angles
is 180(4 – 2) or 360°. Write an equation to express the
sum of the measures of the interior angles of the polygon.
Sum of interior angles
Substitution
Combine like terms.
Subtract 8 from each side.
Divide each side by 32.
Use the value of x to find the measure of each angle.
Answer:
Find the measure of each interior angle.
Answer:
Find the measures of an exterior angle and an interior
angle of convex regular nonagon ABCDEFGHJ.
At each vertex, extend a side to form one exterior angle.
The sum of the measures of the exterior
angles is 360. A convex regular nonagon
has 9 congruent exterior angles.
9e = 360
e = measure of each
exterior angle
e = 40
Divide each side by 9.
Answer: Measure of each exterior angle is 40. Since each
exterior angle and its corresponding interior angle
form a linear pair, the measure of the interior angle is
180 – 40 or 140.
Find the measures of an exterior angle and an interior
angle of convex regular hexagon ABCDEF.
Answer: 60; 120
Polygon Hierarchy
Polygons
Quadrilaterals
Parallelograms
Rectangles
Rhombi
Squares
Kites
Trapezoids
Isosceles
Trapezoids
Quadrilaterals Venn Diagram
Quadrilaterals
Parallelograms
Rhombi
Squares
Rectangles
Trapezoids
Isosceles
Trapezoids
Kites
Quadrilateral Characteristics Summary
Convex Quadrilaterals
Parallelograms
4 sided polygon
4 interior angles sum to 360
4 exterior angles sum to 360
Opposite sides parallel and congruent
Opposite angles congruent
Consecutive angles supplementary
Diagonals bisect each other
Rectangles
Trapezoids
Bases Parallel
Legs are not Parallel
Leg angles are supplementary
Median is parallel to bases
Median = ½ (base + base)
Rhombi
Angles all 90°
Diagonals congruent
All sides congruent
Diagonals perpendicular
Diagonals bisect opposite angles
Squares
Diagonals divide into 4 congruent triangles
Isosceles
Trapezoids
Legs are congruent
Base angle pairs congruent
Diagonals are congruent
Summary & Homework
• Summary:
– If a convex polygon has n sides and sum of the
measures of its interior angles is S, then
S = 180(n-2)°
– The sum of the measures of the exterior angles of a
convex polygon is 360°
– Interior angle + Exterior angle = 180 (linear pair)
• Homework:
– pg 393-95; 1-3, 6-8, 13-15, 17, 22, 28