Unit 3.1
Lesson 2
Introduction to the Angles
1
TABLE OF CONTENTS
Essential Questions…………………………………………………………………..3
Key Definitions and Skills………………………………………………………….4
Who are the angles? ………………………………………………………………..5-7
Classifying angles using a protractor ………………………………………….7-9
Complementary Angles and applications ?……………………………………10-12
Supplementary Angles and Applications? ……………………………………13-15
Vertical and Adjacent Angles and Applications !!!!…………………………16-19
Increasing the rigor with angle concepts……………………………………..20-25
Brain warm up section…………………………………………………………26-27
TOTAL LESSONS:
9
Vocabulary Knowledge Test: January 17, 2013
POST ASSESSMENT: January 22, 2013
UNIT TEST 3.1 :
January 28, 2013
2
Essential Questions
1.
2.
3.
4.
5.
How are angles classified?
What is the difference between complementary and supplementary angles?
What is a unique feature for vertical angles?
How does solving equations help with angles?
What features establish a triangle?
COMMON CORE OBJECTIVES
Cluster:
Solve real-life and mathematical problems involving angle measure, area, surface
area, and volume.
Standard:
Use facts about supplementary, complementary, vertical, and adjacent angles in a
multi-step problem to write and solve simple equations for an unknown angle in a
figure.
Learning Targets:
I can state the relationship between supplementary, complementary, and vertical
angles.
I can use angle relationships to write algebraic equations for unknown angles.
I can use algebraic reasoning and angle relationships to solve multi-step problems
3
KEY DEFINITIONS AND SKILLS
Complementary Angles: are two angles whose measures have a sum of 90⁰
Supplementary Angles: are two angles whose measures have a sum of 180⁰.
Supplementary angles that are adjacent angles from a straight angle.
Vertical Angles: are two non-adjacent angles, formed by intersecting lines, and are
congruent.
Adjacent Angles: are two angles with a side in common.
SKILLS NEEDED
To complete this lesson book, students will need to know how to do the following.
-
Add and subtract two to three digit numbers
Solve two-step equations
Combine like terms
Quickly identify acute, right, and obtuse angles
Write equations
Identify basic inequality signs such as less than and greater than.
4
WHO ARE THE ANGLES?
The Angle Family
Hello! Welcome to the angle to the angle family. We hope you have come for a long
visit. However, while you here we would like to start off by introducing ourselves.
The first of the angle family is the Acute angle.
Acute angles are any angles that have an angle measure of 0 to 89 degrees. They
are the smallest of the angle family.
The next angle family member is the right angle. The right angle is very particular
about the way he looks and does not like change. He is always at 90 degrees.
Being the big brother of the angle family is the obtuse angle. The obtuse angle is
any angle that is 91 degrees to 179 degrees.
5
The last of the angles is the one that got it all figured it out. It decided to take a
straight path. This angle is called a straight angle. This angle measures exactly
180 degrees.
Now that you have been properly introduced let us look at an example.
1. Name one acute angle__________________________
2. Name one obtuse angle_________________________
3. Name one right angle__________________________
4. Name one straight angle_______________________
6
INDEPENDENT PRACTICE
Directions: Use the figure below to answer questions 1 – 4
1. Name all acute angles_______________________________________________
2. Name all obtuse angles______________________________________________
3. Name all right angles________________________________________________
4. Name all straight angles_____________________________________________
_____________________________END OF LESSON______________________________
USING A PROTRACTOR
One way we can construct angles is by using a protractor. Using a protractor
requires four steps.
STEP 1: Draw and label a ray
STEP 2: Line up the center of the protractor on the endpoint of the ray and the 0º
mark of one of scales on the ray.
STEP 3: Mark and label a point where the given measurement is on the same scale
as the 0º mark of step 2.
STEP 4: Use a straight edge to draw a ray from the endpoint of the first ray
through the point that you marked in step 3.
7
EXAMPLE:
Let us use the protractor below to draw an angle that measures 100 degrees.
Notice how we used the top of the protractor not the bottom. This is because we had
to start at 0 degrees and measure up to 100 degrees. The tick marks in between the
numbers represent 1 each.
INDEPENDENT PRACTICE
Use your protractor to construct the angle measure given.
1. 175°
2. 68º
8
3. 32°
4. 127º
INDEPENDENT PRACTICE PART II
Use your protractor to determine the measure of each angle.
1. ____________ degrees
3. _____________ degrees
4. ___________ degrees
2. ______________degrees
9
COMPLEMENTARY ANGLES AND APPLICATIONS
COMPLEMENTARY ANGLES: are two angles whose measures have a sum of 90º
Well welcome back to the angle household. In this house hold, we love to
complement each other. Let me tell you, it is great for business. When the angle
family members complement each, they always add up to be ninety degrees. Let us
look at some examples
LESSON LAUNCH
Now we should know that in a rectangle that the angles are 90 degrees. What if we
were to draw a diagonal that splits the angle? If I told you that one angle is 45
degrees. How much would the other angle be?
EXAMPLE 1
What is the measure of the complement for the angle below?
Focus questions:
What do we know about
complementary angles?
35°
How do we find the angle that
complements∠𝑥𝑦𝑧.
10
EXAMPLE 2
Use the figure below to solve for x.
∠ABD and ∠DBC are complementary angles. What is the value of x?
Focus Questions
1. What does it mean for the angles to be
complementary?
4X
70
2. How will the equation look?
3. How do I solve the equation?
EXAMPLE 3
Name two complementary angles?
___________________________________________________________________________
11
1. What is the measure of the
complement for the angle below?
4. In the diagram below, lines l and m
intersect.
1
3
45°
2
4
________⁰
The measure of ∠4 is 50⁰. Which
equation can be used to find m∠3?
2. Name two complementary angles.
A. m∠3 + 90⁰ = 50⁰
50⁰
60⁰
B. m∠3 - 90⁰ = 50⁰
40⁰
30⁰
C. m∠3 + 50 = 90⁰
D. m∠3 + 50 = 180⁰
_____________________________
5. Angles Y and X are complementary
and m∠Y = 60⁰. What is m∠X?
3. ∠𝐺𝐻𝐽 and ∠𝐷𝐵𝐶 are complementary
A. 20 degrees
angles. What is the value of x?
B. 30 degrees
C. 120 degrees
D. 100 degrees
20⁰
5x + 5
x = _____________
12
SUPPLEMENTARY ANGLES AND APPLICATIONS
Just like we have complementary angles, the angle family can also have
supplement each other. Unlike complementary angles, supplementary angles add
up to be 180⁰.
LESSON LAUNCH
Hockey players want the blades of their sticks to be flat on the ice.
45⁰
Lie Angle
If this player holds his hockey stick to the ice at a 45⁰ angle, at what lie angle
should his stick be? Explain how you know.
EXAMPLE 1
What is the measure of the complement for the angle below?
Focus questions:
What do we know about
supplementary angles?
110°
How do we find the angle that
supplements ∠𝑥𝑦𝑧.
13
EXAMPLE 2
Use the figure below to solve for x.
∠CBD and ∠ABD are supplementary angles. What is the value of x?
Focus Questions
1. What does it mean for the angles to be
supplementary angles?
4X
70
2. How will the equation look?
3. How do I solve the equation?
EXAMPLE 3
Name two supplementary angles?
___________________________________________________________________________
14
1. What is the measure of the
supplement of the angle below?
4. . Use the figure below to answer
the question.
168⁰
x⁰
(x + 30)⁰
_________
If angle ABC and angle CBE are
supplementary, then what equation
can be used to find the value of angle
ABC?
2. Suppose m∠PQR = 63.7⁰. Find the
measure of its supplement.
A. 3x + 30 = 90
B. 2x + 30 = 90
C. x² + 30 = 180
D. 2x + 30= 180
5. In the diagram below, m∠2 = 90⁰.
____________
3. What is the value of x?
Which statement must be true?
A. ∠2 and ∠5 are vertical
4x -2⁰
B. ∠3 and ∠4 are congruent
46⁰
C. ∠1 and ∠3 are complementary
D. ∠2, ∠3, and ∠4 are supplementary.
x = _____________
15
VERTICAL AND ADJACENT ANGLES AND APPLICATIONS
EXAMPLE 2
Vertical Angles
Vertical angles are CONGRUENT,
that is the bottom line. They are
located across from each other and
share the same vertex point.
2
EXAMPLE
3
1
4
Another application that can be
applied using vertical angles is solving
for x.
Let us say that m∠1 = 5x + 20⁰ and
m∠3 = 50⁰.
By looking at the figure above, we can
see that ∠ ABC and ∠DBE are vertical
angles, which means they are
congruent. They also share a common
vertex B.
STEP 1: Since vertical angles are
congruent we can set angle 1 and
angle 3 equal to each other.
5x + 20 = 50
Let us see if we can name another set
of vertical angles from the figure
above.
STEP 2: Now that we have an
equation, solve using the “T” method.
∠__________ & ∠_____________
5x + 20
-20
______________
5x ÷ 5
50
-20
___________
30 ÷ 5
x
The value of x equals to 6.
16
6
Adjacent Angles
EXAMPLE
Just like back in complementary and
supplementary angles. We saw that
the angles were sometimes right next
to each other. It was almost as if they
were next door neighbors. That is
exactly what adjacent angles are.
They are angles that share a common
ray. Let us look at an example.
EXAMPLE
Let us look at the figure above to
name 3 pair of adjacent angles. There
are several different angles
_______________________
In the example above, we can see that
∠CBA and ∠DBA are adjacent. This is
because these two angles share line
BA together.
_______________________
In many cases, we will see that
adjacent angles occur when angles are
split into two angle and angles form
on a straight line.
Now let us complete some practice
using vertical and adjacent angles
_______________________
The great thing about adjacent angles
is that they can help us solve for many
unknown angles. So you thank them
later. Let us look at one more example
before we move on.
17
VERTICAL ANGLES
18
ADJACENT ANGLES
19
INCREASING THE RIGOR USING ANGLE CONCEPTS
LET US SOLVE IT OUT
20
21
22
A. 42
B. 36
C. 33
D. 22
23
24
25
BRAIN WARM UP
Please complete the one assigned for each day. If there is not one, that means that
we have a quiz or other event.
MONDAY
A rectangular room on a blueprint is 2 inches wide and 5 inches long. If the scale of
the blue print to the actual room is 1:72, what is the width of the room?
A. 12ft
B. 30ft
C. 144ft
D. 360ft
TUESDAY
26
WEDNESDAY
FRIDAY
27