Download This week, we will learn how to find the area and angles of regular

Name:
Date:
Period:
This week, we will learn how to find the area and angles of regular polygons. We will end the week reviewing how to solve proportions.
CLASS
MONDAY (9/19)
Homework
2nd Hour
1st Hour
Practice with finding the
apothem
TUESDAY (9/20)
Review Angles
WEDNESDAY (9/21)
THURSDAY (9/22)
FRIDAY (9/23)
Interior Angles of a polygon
More practice with angles
& polygons
Review game
Angles in a Triangle
Area of regular polygons
Practice with angles in a
triangle
Angles in a regular polygon
MDTP
Quiz/ALEKS
Do page 6.
Watch the video on angles.
Take notes and do page 78.
p. 15-17
p. 25-26
Study for your quiz
tomorrow. Do p. 31-32 to
help you review.
Watch video on how to
solve proportions. Take
notes. Do p. 33.
1
Practice with Finding the Height of a Triangle/Apothem
Do these problems on a separate piece of paper!
2
Area of Polygons
In the previous lesson, you learned how to find the height and area of certain triangles using the Pythagorean Theorem.
Now, you will learn how to find the area of regular polygons like a pentagon or a decagon! Today you will explore these
different polygons and generalize how to find the area of any regular polygon with n sides.
1.) FIND THE AREA OF THESE POLYGONS
With your team, find the area of each shape below. Be sure that each member of your team understands your methods.
You may be asked to present your methods in front of the class.
The radius of this hexagon is also
8. Look below to see what radius
means.
3
2.) Using the methods that your classmates presented in part 1, find the area of the following polygons.
Radius = 5.3; Apothem = 5
7.5
3.)
The radius
is 12.8 ft.
4
4.)
Each side of the octagon
is equal to 1.53 ft.
Journal: Explain how you generally find the area of a polygon. How would you find the area of a regular polygon with n
sides if you were given the length of the apothem and the length of a side?
5
Homework
Try your best on these problems. They may be difficult. Try your best and we’ll go over them in class.
6
Introduction to Angles
1. Name each angle in 3 different ways
a.
b.
2. Draw and label each angle
a.
b.
3. For the figure below, list the angles that you can name using only the vertex letter.
4. Find the measure of each of the following angles.
a.) m AQB = _______
b.) m ZOY = _______
c.) m AQC = _______
d.) m ZQX = _______
e.) m XQA= _______
f.) m CQB = _______
g.) m AQY = _______
h.) m XQY = _______
7
A.
B.
C.
8
Practice with Angles
Angle Relationship Practice
Remember:
 Congruent means equal measures
 Complementary means that the sum of two angles is 90 degrees and supplementary means that the sum of two
angles is 180 degrees
 Linear Pairs are two angles that share a vertex and a common side and their noncommon side forms a line
 Vertical Angles are angles formed by 2 intersecting lines that share a vertex but do not have a common side.
D.
E.
9
Angles in a Triangle
Today we are going to study the angles in a triangle.
1.) How many degrees is a straight line? _____________
2.) Marcos decided to study angle relationships in triangles by drawing a pattern. Here it is:
a.) Trace one of the triangles onto a piece of paper. Use one of the colored pen or pencil to shade one
of the triangle’s angles onto the tracing paper. Then use the same color to shade every angle on the
pattern above that is equal to the shaded angle.
b.) Repeat this process for the other two angles of the triangle.
c.) Examine the coloring pattern. What relationship can you find between the three different colored-angles? Focus on
the angles that form a straight angle. What does this tell you about the angles in a triangle?
d.) Triangle Sum DEMO. http://www.cpm.org/flash/technology/triangleSum.swf
What did the triangle sum demo tell you about the angles in a triangle?
The sum of the angles in a triangle = ___________
10
3.) Use what you know about angles in a triangle to find x in each diagram below:
4.) Find the angles of each of these special triangles:
a.) Equilateral Triangle
b.) Isosceles Right Triangle
c.) What if you only know one angle of an isosceles triangle? For example, if m∠A = 34°, what are
the measures of the other two angles?
11
Practice with Triangles
A. Match the term with the picture on the right.
1. Equilateral Triangle
2. Scalene right triangle
3. Isosceles right triangle
4. Isosceles obtuse triangle
B. Sketch and label the figure. Mark the picture you draw with symbols to indicate congruent sides and angles.
1. Isosceles acute triangle ACT with AC = CT
2. Scalene triangle SCL with angle bisector segment CM.
3. Isosceles right triangle CAR with mCRA = 90.
C. Use the graph below to help you answer the question.
1. Where can point L be so that ∆LRY is an isosceles triangle?
2. Where should point O be so that ∆MOE is an isosceles right triangle.
12
D.
E.
F.
13
G.
14
Homework
15
16
Warmup
17
Interior Angles in a Polygon
In a previous lesson, you discovered that the sum of the interior angles of a triangle is always 180°. But what about the
sum of the interior angles of other polygons, such as hexagons or decagons?
1.) Examine the geometric relationships in the diagram at right. Show all the
steps in your solutions for x and y.
2.) Look at the diagram of the regular pentagon on the right. With your team, find the sum of the measures of its
interior angles as many ways as you can. You may want to use the fact that the sum of the angles of a triangle is 180°. Be
prepared to share your methods.
Journal about the method(s) that you used to find the sum of angles in the pentagon. Do you think your method would
work for a pentagon that is NOT regular?
3.) The pentagon at right has been dissected (broken up) into three triangles with the angles labeled as shown. Use the
three triangles to prove that the sum of the interior angles of any pentagon is always 540°.
a.) What is the sum of the angles of a triangle? Use this fact to write three equations
based on the triangles in the diagram.
b.) Add the three equations to create one long equation that represents the sum of all
nine angles.
18
4.) SUM OF THE INTERIOR ANGLES OF A POLYGON
In the first part of the lesson, you found the sum of the angles of a regular pentagon. But what about other polygons?
19
5.) Find the sum of the interior angles of a 100-gon. Explain your reasoning.
6.) Use the angle relationships in each of the diagrams below to solve for the given variables. Show all work.
20
Angles of Regular Polygons
We just discovered how to find the sum of the interior angles of a polygon with any number of sides. But what more can
you learn about a polygon? Today you will focus on the interior and exterior angles of regular
polygons.
1.) Diamonds, the most valuable naturally-occurring gem, have been popular for centuries
because of their beauty, durability, and ability to reflect a spectrum of light. In 1919, a
diamond cutter from Belgium, Marcel Tolkowsky, used his knowledge of geometry to design a
new shape for a diamond called the “round brilliant cut” (top view shown at right). He
discovered that when diamonds are carefully cut with flat surfaces (called “facets” or “faces”)
in this design, the angles maximize the brilliance and reflective quality of the gem.
Notice that at the center of this design is a regular octagon with equal sides and equal interior
angles. For a diamond cut in this design to achieve its maximum value, the octagon must be cut
carefully and accurately. One miscalculation, and the value of the diamond can be cut in half!
a.) Determine the measure of each interior angle of a regular octagon. Explain how you found
your answer.
b.) What about the interior angles of other regular polygons? Find the interior angles of a regular nonagon and a regular
100-gon.
c.) Will the process you used for part (a) work for any regular polygon? Write an expression that will calculate the
interior angle of an n-gon.
21
2.) Fern states, “If a triangle is equilateral, then all angles have equal measure and it must be a regular polygon.” Does
this logic work for polygons with more than three sides? Is Fern’s statement correct for all types of regular polygons?
a.) If all the sides of a polygon (such as a quadrilateral) are equal, does that mean that the angles must be equal? If you
can, draw a counterexample (draw a polygon that proves the statement wrong).
b.) What if all the angles are equal? Does that force a polygon to be equilateral? Explain your thinking. Draw a
counterexample on your paper if possible.
3.) Jeremy asks, “What about exterior angles? What can we learn about them?”
a.) Examine the regular hexagon shown at right. Angle a is an example of an exterior
angle because it is formed on the outside of the hexagon by extending one of its sides.
Are all of the exterior angles of a regular polygon equal? Explain how you know.
b.) Find a. Be prepared to share how you found your answer.
c.) This regular hexagon has six exterior angles, as shown in the diagram above. What is the sum of the exterior angles of
a regular hexagon?
22
d.) What about the exterior angles of other regular polygons? Explore this with your team. Have each team member
choose a different shape from the list below to analyze. For each shape:


Find the measure of one exterior angle of that shape
Find the sum of the exterior angles
Shape
Diagram/Measure of one interior angle
Measure of one exterior
angle
Sum of the exterior angles
1.) equilateral
triangle
2.) regular
octagon
3.) regular
decagon
4.) regular
dodecagon (12gon)
e.) Compare your results from part (d). As a team, write a statement about the exterior angles of polygons based on
your observations.
f.) Is this statement from part (e) true for all polygons or for only regular polygons? Explore these questions with a demo
http://www.cpm.org/flash/technology/externalangles.swf. Write a statement explaining your findings.
23
4.) Use your understanding of polygons to answer the questions below, if possible. If there is no solution, explain why
not.
a.) Gerado drew a regular polygon that had exterior angles measuring 40°. How many sides did his polygon have? What
is the name for this polygon?
b.) A polygon has an interior angle sum of 2,520°. How many sides does it have?
c.) A quadrilateral has four sides. What is the measure of each of its interior angles?
d.) What is the measure of an interior angle of a regular 360-gon? Is there more than one way to find this answer?
24
Homework
1.) How can you find the interior angle of a regular polygon?
2.) What is the sum of the exterior angles of a polygon?
25
26
Angles in Polygons
A.
B.
Find the measure of A
C.
Find the measure of an interior angle of the regular polygon.
27
D.
Find the value of x.
E.
28
More Practice with Angles & Polygons
During the previous lessons this week, you have discovered many ways the number of sides
of a regular polygon is related to the measures of the interior and exterior angles of the
polygon. These relationships can be represented in the diagram to the right.
1.) What connections in the Polygon Angle Web do you have already? Which do you still
need? Explore this as you answer the questions below.
a.) If you know the number of sides of a regular polygon, how can you find the measure of an interior angle directly?
Find the measurements of an interior angle of a 15-gon.
b.) If you know the number of sides of a regular polygon, how can you find the measure of an exterior angle directly?
Find the measurements of an exterior angle of a 10-gon.
c.) What if you know that the measure of an interior angle of a regular polygon is 162°? How many sides must the
polygon have? Show all work.
d.) If the measure of an exterior angle of a regular polygon is 15°, how many sides does it have? What is the measure of
an interior angle? Show how you know.
29
2.) Suppose a regular polygon has an interior angle measuring 120°. Find the number of sides using two different
strategies. Show all work. Which method was most efficient?
Method #1:
Method #2:
3.) Use your knowledge of polygons to answer the questions below, if possible.
a.) How many sides does a polygon have if the sum of the measures of the interior angles is 1980°? 900°?
b.) If the exterior angle of a regular polygon is 90°, how many sides does it have? What is another name for this shape?
c.) Each interior angle of a regular pentagon has the measure 2x + 4°. What is x? Explain how you found your answer.
d.) The measures of four of the exterior angles of a pentagon are 57°, 74°, 56° and 66°. What is the measure of the
remaining angle?
e.) Find the sum of the interior angles of an 11-gon. Does it matter if it is regular or not?
30
Homework
1.) The exterior angle of a regular polygon is 20°.
a.) What is the measure of an interior angle of this polygon. Show how you know.
b.) How many sides does this polygon have? Show all your work.
2.) Find the area of an equilateral triangle with side length 20 mm. Draw a diagram and show all work.
3.) Solve for x in each diagram below.
x
31
4.) What is another (more descriptive) name for each polygon described below?
a.) A regular polygon with an exterior angle measure 120°.
b.) A quadrilateral with four equal angles.
c.) A polygon with an interior angle sum of 1260°.
32
Practice with Solving Proportions
33