Lesson 1-6
Two-Dimensional Figures
Lesson Outline
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Five-Minute Check

Then & Now and Objectives
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Vocabulary
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Key Concept
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Examples
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Lesson Checkpoints
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Summary and Homework
Then and Now
You measured and classified angles.
• Identify and use special pairs of angles
• Identify perpendicular lines
Objectives
• Identify and use special pairs of angles
• Identify perpendicular lines
Vocabulary
• Adjacent angles – two coplanar angles that have a common vertex,
a common side, but no common interior points
• Linear pair – a pair of adjacent angles whose noncommon sides
are opposite rays (always supplementary)
• Vertical angles – two non adjacent angles formed by two
intersecting lines
Vertical angles are congruent (measures are equal)!!
• Complementary Angles – two angles whose measures sum to 90°
• Supplementary Angles – two angles whose measures sum to 180°
• Perpendicular – two lines or rays are perpendicular if the angle (s)
formed measure 90°
Key Concept
• Closed figure (no escape)
• sides are all line segments (no curves)
Not a Polygon
Figure is not closed
Sides are not
line segments
Polygons
Side
extended
goes
through
interior
Concave
Convex
Not Concave
All extended sides
stay outside interior
Interior Angle
> 180°
All Interior Angles
less than 180°
Irregular
Not Regular
Regular
All Sides same
All Angles same
Perimeter
P=a+b+c+d+e+f
e
f
d
Once around
the figure
a
c
b
If regular,
then a = b = c = d = e = f
and P = 6a
Names of Polygons
Number
of Sides
Name
Sum of
Interior
Angles
3
4
Triangle
Quadrilateral
180
360
5
6
7
Pentagon
Hexagon
Heptagon
540
720
900
8
9
Octagon
Nonagon
1080
1260
10
12
n
Decagon
Dodecagon
N-gon
1440
1800
(n-2) • 180
Example 1A
Name the polygon by its number of sides. Then
classify it as convex or concave, regular or irregular.
There are 4 sides, so this is a quadrilateral.
No line containing any of the sides will pass through the
interior of the quadrilateral, so it is convex.
The sides are not congruent, so it is irregular.
Answer: quadrilateral, convex, irregular
Example 1B
Name the polygon by its number of sides. Then
classify it as convex or concave, regular or irregular.
There are 9 sides, so this is a nonagon.
A line containing some of the sides will pass through the
interior of the nonagon, so it is concave.
The sides are not congruent, so it is irregular.
Answer: nonagon, concave, irregular
Key Concept
• Perimeter, P: once around the outside
• Area, A: square units of measure
Example 2A
A. Find the perimeter and area of the figure.
P = 2ℓ + 2w
Perimeter of a rectangle
= 2(4.6) + 2(2.3)
ℓ = 4.6, w = 2.3
= 13.8
Simplify.
Answer: The perimeter of the rectangle is 13.8 cm.
Example 2A cont
A. Find the perimeter and area of the figure.
A = ℓw
Area of a rectangle
= (4.6)(2.3)
ℓ = 4.6, w = 2.3
= 10.58
Simplify.
Answer: The area of the rectangle is about 10.6 cm2.
Example 2B
B. Find the circumference and area of the figure.
C = 2r
C = 2(4)
≈ 25.1
Circumference of a circle
r=4
Use a calculator.
Answer: The circumference of the circle is about
25.1 inches.
Example 2B cont
B. Find the circumference and area of the figure.
A = r2
A = (4)2
≈ 50.3
Area of a circle
r=4
Use a calculator.
Answer: The area of the circle is about 50.3 square
inches.
Example 3
Terri has 19 feet of tape to mark an area in the classroom
where the students may read. Which of these shapes
has a perimeter or circumference that would use most or
all of the tape?
A square with side length of 5 feet
B circle with the radius of 3 feet
C rectangle with a length of 8 feet and a width of 3 feet
D right triangle with each leg length of 6 feet
Example 3 cont
Solve the Test Item
Find each perimeter or circumference.
Square
P = 4s
= 4(5) = 20 ft
Circle
C = 2r
Perimeter of a square
s = 5 and Simplify
Circumference
= 2(3) = 6 ≈ 18.85 ft
r = 3 and Simplify
Rectangle
P = 2ℓ + 2w
= 2(8) + 2(3) = 22 ft
Perimeter of a rectangle
ℓ = 8, w = 3 and Simplify
Example 3 cont
Right Triangle
Use the Pythagorean Theorem to find the length of the
hypotenuse.
c2 = a2 + b2
Pythagorean Theorem
= 62 + 62
a = 6, b = 6
= 72
Simplify.
Take the square root of each side
Use a calculator.
≈ 8.49
P =a+b+c
Perimeter of a triangle
 6 + 6 + 8.49
Substitution
 20.49 feet
Simplify.
The only shape for which Terri has enough tape is the circle.
Answer: The correct answer is B.
Example 4
Find the perimeter and area of a pentagon ABCDE
with A(0, 4), B(4, 0), C(3, –4), D(–3, –4), and E(–3, 1).
Step 1a
By counting squares on the
grid, we find that CD = 6
units and DE = 5 units.
Example 4 cont
Step 1b
Use the Distance Formula,
to find AB, BC, and EA.
The perimeter of pentagon ABCDE is
5.7 + 4.1 + 6 + 5 + 4.2 or about 25 units.
Example 4 cont
Step 2
Divide the pentagon into two
triangles and a rectangle.
Find the area of the triangles.
Area of Triangle 1
Area of a triangle
Area of Triangle 2
Substitute.
Substitute.
Simplify.
A = 2.5
Simplify.
Example 4 cont
Step 2 continued
Find the area of the rectangle.
A = lw
Area of a rectangle
A = (5)(6)
Substitute.
A = 30
Simplify.
The area of pentagon ABCDE is
9 + 2.5 + 30 or 41.5 square units.
Answer: The perimeter is about 25 units and the
area is 41.5 square units.
Lesson Checkpoints
Summary & Homework
• Summary:
– A polygon is a closed figure made of line segments
– The perimeter of a polygon is the sum of the lengths
of its sides
– The perimeter of a circle is called the circumference
– The area of a two-dimensional figure can be
determined by formulas on the formula sheet
• Homework:
– pg 61-3: 11-16, 18-21, 34-36