Download Lesson 4.2: Angles In a Polygon

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Geometry Unit 4: Polygons – Student Guide
Lesson Title
Content Objectives
Language Objectives
Common
Core
Standards
DP Competencies
Time
(Days)
4.1 Classifying
Polygons
*Remedial
SWBAT classify
polygons and indicate
if they are regular or
not.
G-CO,10
M04
1
4.2: Angles in a
Polygon
SWBAT use the
Polygon Interior
Angles Theorem and
Polygon Exterior
Angles theorem to
calculate missing
angle measures.
SWBAT calculate the
area and perimeter of
a triangle.
SWBAT explain the
difference between
equilateral,
equiangular, and
regular polygons
through using a Venn
Diagram.
SWBAT explain how to
calculate the interior
angles of a polygon
when given number of
angles or sides.
G-CO,10
M01
M04
2
SWBAT explain in
writing how to find the
area and perimeter of
a triangle.
SWBAT explain how to
find the area of a
parallelogram,
trapezoid, and kite in
writing using triangles.
G-CO,10
G-MG.3
M01
1
G-CO.11
G-MG.3
M01
M04
3
M01
1
4.3: Triangles
4.5:
Quadrilaterals
4.6: More on
Polygons
1
SWBAT apply
different properties of
special quadrilaterals
and use them to
calculate the area and
perimeter of them.
SWBAT calculate the
area and perimeter of
regular polygons
SWBAT explain how to G-MG.3
find the area of a
regular polygon in
writing using triangles.
Lesson 4.1: Classifying Polygons.
Content Objective: SWBAT classify polygons and indicate if they are regular or not.
Language Objective: SWBAT explain the difference between equilateral, equiangular, and regular
polygons through using a Venn Diagram.
Read pages 303-305 of Holt McDougal’s Geometry book and
answer the following:
1.
2.
3.
4.
What are characteristics of a polygon?
What is a diagonal?
How many diagonals do a 5-sided polygon have?
Complete the table in your notebook (draw a picture
and identify the number of sides)
Triangle
Hexagon
Quadrilateral
Heptagon
Pentagon
Octagon
Practice:
Problems 1-2, 8-14 on page 306 of Holt McDougal’s Geometry book
Read pages 411 and 412 of Holt McDougal’s
Geometry book and draw a Venn diagram below
to compare equilateral, equiangular, and
regular polygons.
Practice:
Problems 7, 13-21, 27-32 on pages 413-414 of Holt McDougal’s Geometry book
1. How can you tell if polygons are regular?
Lesson 4.2: Angles In a Polygon
Content Objective: SWBAT calculate missing interior and exterior angle measures and find unknown
variables using the Polygon Sum conjecture.
Language Objective: SWBAT explain how to calculate the interior angles of a polygon when given number
of angles or sides.
2
Use the diagram on the top of page 417 of Holt McDougal’s Geometry book to identify the difference
between interior angels and exterior angles on a polygon and explain it in your notebook.
Interior:
Exterior:
Calculating Interior Angles:
Complete the activity on page 416 of Holt McDougal’s Geometry book. You can use the polygons below to
draw diagonals and fill out the table below:
Polygon Name
Number
of Sides
Number
of
triangles
Sum of measures of interior angles
(To find this multiply 180 by the number of triangles)
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
N-gon
n
Write down the words and symbols of Theorem 8.1, the Polygon Interior Angles Theorem on page 417 of
Holt McDougal’s Geometry book in your notebook.
Read examples 1-3 and answer the following:
1) Why did the author substitute 7 for n in example 1?
2) Why did the author set the equation equal to 720° in example 2?
3) Will every regular octagon have interior angles that measure 135°?
Calculating Exterior Angles:
Calculating exterior angles is much easier. All exterior angles in a polygon (number of sides doesn’t
matter) add up to 360°.
3
Write down the words and symbols of Theorem 8.2, the Polygon Exterior Angles Theorem on page 419 of
Holt McDougal’s Geometry book in your notebook.
Read example 4 and answer the following:
4) Why did the author set the equation equal to 360° in example 4?
Practice:
Problems 4-7 (all), 8-16 (evens), 17-28 (all) on pages 420-421 of Holt McDougal’s Geometry book AND
the Checkpoint at the top of page 420.
Higher Level Thinking Questions:
11) If the sum of the measures of the interior angles of a polygon is twice the sum of its exterior angles,
how many sides does it have? Explain your thinking.
12) How many sides does a regular polygon have if each exterior angle measures 30°? Explain your
thinking.
13) Explain in your own words how to calculate the sum of the interior angels in a polygon. Be sure to use
the words diagonals and triangles in your description.
Lesson 4.3: Triangles
Content Objective: SWBAT calculate the area and perimeter of a triangle.
Language Objective: SWBAT explain in writing how to find the area and perimeter of a triangle.
Read the following:
Area: The amount of surface the 2D shape covers. It’s
measured in square units. Some of these units are:
squared centimeters (cm2), squared meters (m2) and
squared kilometers (km2).
The general way to calculate the area of a rectangle is
by multiplying base by height (altitude). However, to
calculate the area of triangle, you must use either of the
following equations:
*An easy way to remember this is that a triangle is
exactly half the size of a rectangle!
Perimeter: The total distance around the outside of a 2D shape. You calculate it by adding together all of
the lengths of the shape.
In the triangle above, you can use the following formula to calculate the perimeter:
4
Complete the following table in your notebook using the information you just read and learned in
previous lessons:
Triangles
Properties of Triangles
Definition
Picture
Sum of the Interior Angles
Sum of the Exterior Angles
Isosceles Triangle
Equilateral/Regular Triangle
Area
Perimeter
Practice:
Problems 1, 2, 5, 6, 10, 12 on page 430-431 of Discovering Geometry.
Literacy Questions:
Answer the following in your notebook:
1. Some students try to use side “c” as the height instead of the altitude (h). Explain why this doesn’t
work. Use what you know about altitude from Unit 2 in your explanation.
Lesson 4.4: Quadrilaterals
Content Objective: SWBAT apply different properties of special quadrilaterals and use them to calculate
the area and perimeter of them.
Language Objective: SWBAT explain how to find the area of a parallelogram, trapezoid, and kite in writing
using triangles.
A Quadrilateral is a polygon with four sides. Below is a diagram of the quadrilateral family.
5
6
Rectangles, Squares, Parallelograms, Rhombus
Properties: To learn the properties of these shapes, consult the diagram on the previous page and answer
the following questions:
1.
2.
3.
4.
Are all squares rectangles? Are all rectangles squares? Explain your reasoning.
What is different between the diagonals of parallelograms and the diagonals of rhombuses?
How is a rhombus similar to a square?
In a parallelogram, what does “Consecutive angle pairs supplementary” mean?
Practice:
Complete problems 1-6 on page 283 of Discovering Geometry.
Calculating the area of each shape:
Rectangle’s and Squares
Parallelograms and Rhombuses
To calculate the area of either a rectangle or
square, simply multiply base x height.
To calculate the area of either a parallelogram or
rhombus, simply multiply base x height.
𝐴 = 𝑏ℎ
*It is important your height is perpendicular to the
base.
𝐴 = 𝑏ℎ
Practice:
Complete problems 1-10, 17, 18 on pages 425-426 of Discovering Geometry.
Kites and Trapezoids:
Properties: To learn the properties of these shapes, consult the diagram on the previous page and answer
the following questions:
1. How is a rhombus different from a kite?
2. How is a trapezoid different from a parallelogram? Can a trapezoid ever be a parallelogram?
3. Which pair of opposite angles are congruent in a kite?
4. What does “uncommon base angles supplementary” mean in an isosceles trapezoid?
7
Practice:
Complete problems 1-6 on page 271 of Discovering Geometry.
Calculating the area of each shape:
Kites
Trapezoids
To calculate the area of a kite, multiply the two
diagonals together and divide them by 2.
To calculate the area of a trapezoid, multiply the
average of the bases by the height.
𝐴=
𝑏ℎ
2
𝐴=
𝑏1 + 𝑏2
ℎ
2
Practice:
Complete problems 3, 4, 7, 8, 9, 11, 14, 15 on pages 430-431 of Discovering Geometry.
Literacy:
Answer the following in your math notebook:
1. Explain why it is important to find the average of the bases when finding the area of a trapezoid.
2. Explain why you must divide by 2 when finding the area of a kite.
3. Using what you know about triangles, explain how to find the area of a parallelogram.
8
Lesson 4.6: More on Polygons
Content Objective: SWBAT calculate the area and perimeter of regular polygons
Language Objective: SWBAT explain how to find the area of a regular polygon in writing using triangles.
We will now expand our knowledge of polygons to shapes with more than 4 sides.
Regular Polygons
…
5 sides
6 sides
7 sides
8 sides
Irregular Polygons
Explain the difference between a regular polygon and an irregular polygon.
Sides:
Angles:
Area of a Regular Polygon: To find the area of a regular polygon we can break up the shape up into
congruent triangles.
Area = Area of Triangle  Number of Triangles
3 in
4 in
 Base  Height 
Area = 
 N
2


 3 4 
2
Area = 
  6  36 in
 2 
The height of the triangle in the regular polygon is also known
9
at
the apothem. The apothem measures the distance from the center of the polygon to the middle of a side.
Practice:
Complete the problems below AND problems 1-8 on page 443 of Discovering Geometry.
Find the area and perimeter of each regular polygon. Identify the name of the polygon and draw the
triangles to help you find the perimeter.
Literacy:
Answer the following in your math notebook:
1. Adapted from Discovering Geometry: A square is also a regular polygon. How is the apothem of a
square related to the side length? Show that that the formula used to find the area of a regular
polygon simplifies to s2 for the area of a square.
10