Download ALL ABOUT ANGLES

ALL ABOUT ANGLES
INTRODUCTION
The objective for this lesson on All About Angles is, the student will use facts about
supplementary, complementary, vertical and adjacent angles to solve mathematical and
real-world problems.
The skills students should have in order to help them in this lesson include, classifying
angles, (acute, obtuse, right and straight) and solving equations.
We will have three essential questions that will be guiding our lesson. Number one,
explain the meaning of complementary angles and supplementary angles. Number two,
Explain the meaning of adjacent and vertical angles. And number three, explain how you
can use angle relationships and triangle relationships to find the missing angle in a
triangle.
Begin by completing the warm-up on identifying angles to prepare for the lesson called
all about angles.
SOLVE PROBLEM – INTRODUCTION
The SOLVE problem for this lesson is, Samantha is designing a triangular garden bed in
a new city park. The drawing below shows where the garden bed meets a walking path.
What is the measurement of Angle c?
In Step S, we Study the Problem. First we need to identify where the question is located
within the problem and underline the question. The question for this problem is, what is
the measurement of Angle c?
Now that we have identified the question, we need to put this question in our own words
in the form of a statement, this problem is asking me to find the angle measurement of
Angle c.
During this lesson we will learn about angle relationships in order to complete this solve
problem at the end of the lesson.
ANGLE RELATIONSHIPS
We will be looking at four different types of angle relationships. There are four types of
angle relationships listed in the word bank. Vertical, Supplementary, Complementary
and Adjacent.
Let’s measure each pair of angles in Problem One.
ANGLE RELATIONSHIPS – COMPLEMENTARY ANGLES
Set A has two angles we’re going to measure. The first angle measures twenty three
degrees. The second angle measures sixty seven degrees.
In Set B, the first angle measures forty eight degrees. The second angle measures forty
two degrees.
Set C, the first angle measures twenty three degrees. The second angle measures forty
eight degrees.
Set D, the first angle measures sixty seven degrees. The second angle measures forty two
degrees.
Let’s fill in the chart with the measures of the first angle and the second angle for each
angle pair.
Angle pair A, the first angle was twenty three degrees, the second angle was sixty seven
degrees. What is the sum of the two angles for Angle Pair A? Ninety Degrees
Angle pair B, the first angle was forty eight degrees, the second angles was forty two
degrees. What is the sum of the two angles for Angle Pair B? Ninety Degrees
Angle pair C, the first angle had a measure of twenty three degrees, and the second angle
had a measure of forty eight degrees. What is the sum of the two angles for Angle Pair
C? Seventy One Degrees
Angle pair D, the first angle measured sixty seven degrees, and the second angle
measured forty two degrees. What is the sum of the two angles for Angle Pair D? One
hundred nine degrees
Using a ruler and protractor, let’s draw each pair of angles from the previous page using a
common ray. Set A, Set B, Set C, and Set D.
What do you notice about the angle that is formed by the two angles in One A and One
B? In both Set A and Set B the two angles have a sum of ninety degrees and form right
angles.
What do you notice about the angle that is formed by the two angles in One C and One
D? These angle pairs do not form right angles, because they do not have a sum of ninety
degrees.
Our Conclusion is this: When two angles add up to ninety degrees, we say they
complement each other. When two angles add up to ninety degrees, they are
complementary angles.
ANGLES RELATIONSHIPS – SUPPLEMENTARY ANGLES
Use your protractor to measure each angle for Problem Four on the bottom of the page.
Sets C and D continue on the top of the next page.
Set A – the first angle measures one hundred degrees. The second angle measures eighty
degrees.
Set B – the first angle measures one hundred twenty degrees. The second angle measures
sixty degrees.
Set C – the first angle measures one hundred degrees. The second angle measures sixty
degrees.
And Set D – the first angle measures one hundred twenty degrees. The second angle
measures eighty degrees.
Let’s fill in the chart with the measures of the first angle and the second angle for each
angle pair.
Angle Pair A, our first angle measures one hundred degrees. Our second angle measures
eighty degrees. What is the sum of the two angles for Angle Pair A? One hundred eighty
degrees
Angle Pair B, the first angle measures sixty degrees. The second angle measures one
hundred twenty degrees. What is the sum of the two angles for Angle Pair B? One
hundred eighty degrees.
Angle Pair C, the first angle has a measure of one hundred degrees. The second angle
has a measure of sixty degrees. What is the sum of the two angles for Angle Pair C? The
sum is one hundred sixty degrees.
Angle Pair D, the first angle has a measure of one hundred twenty degrees. The second
angle has a measure of eighty degrees. What is the sum of the two angles for Angle Pair
D? Two hundred degrees
Using a ruler and protractor, let’s draw each pair of angles from the previous page using a
common ray.
Set A – Our two angles measure one hundred and eighty degrees.
Set B – Our two angles measure sixty degrees and one hundred twenty degrees.
Set C – Our two angles measure one hundred degrees and sixty degrees.
And Set D – Our two angles measure one hundred twenty degrees and eighty degrees.
What do you notice about the angle that is formed by the two angles in Four A and Four
B? The two angles in Set A, and the two angles in Set B, have a sum of one hundred
eighty degrees and form a straight line.
What do you notice about the angle that is formed by the two angles in Set Four C and
Four D? These angle pairs do not form a straight line.
Conclusion: When two angles add up to one hundred eighty degrees, we say they
supplement each other. When two angles add up to one hundred eighty degrees, they are
supplementary angles.
ANGLE RELATIONSHIPS – VERTICAL ANGLES
Use a protractor to measure the angles in Problem Seven.
In Figure A, angle a measures forty five degrees. Angle b measures one hundred thirty
five degrees. Angle c measures forty five degrees. Angle d measures one hundred thirty
five degrees.
Let’s look at Figure B, angle a measures sixty degrees and angle b measures one hundred
twenty degrees. Angle c measures sixty degrees and angle d measure one hundred
twenty degrees.
What do you notice about the measurements in Figure A? Angle a is congruent to Angle
c. Angle b is congruent to Angle d.
What do you notice about the measurements in Figure B? Angle a is congruent to Angle
c. Angle b is congruent to Angle d.
Conclusion: When two lines intersect, they form four angles. The angles that are
opposite each other are congruent and are known as vertical angles.
ANGLE RELATIONSHIPS – ADJACENT ANGLES
Let’s look at Figure A, Figure B, and Figure C. What do you notice about each pair of
angles? They are connected and share a ray with a common endpoint.
What do we call angles that share a ray? Angles that share a ray are called adjacent
angles.
ANGLE RELATIONSHIPS IN TRIANGLES
Draw a triangle on a separate sheet of paper. It does not matter what type of triangle, but
it should be large enough to cut out. Use a ruler or the side of the protractor to be sure
the sides are straight.
Color each of the angles a different color using colored pencils.
Measure all three angles and place the measurements in the table. Record the
measurements of your partner’s triangle also.
Are any of the triangles exactly alike? We most likely all have different triangles!
How many different triangles do we have? That would depend on the number of
students.
Are there any acute triangles? Obtuse triangles? Right? Isosceles? Scalene?
Equilateral?
Take a look at Problem Five.
What is the sum of the angles of your triangle? One hundred eighty degrees
What is the sum of the angles in your partner’s triangle? One hundred eighty degrees
Now, let’s tear the triangle into three separate angles. You have two different options.
The first option is to tear off the points of the triangles, such that you tear off three
angles, as shown with the dashed lines above.
The second option is to tear so that three pieces of the triangle are created.
The first option will be displayed. Now that the pieces are cut apart, place the three
angles on the line.
Line up the angles so the lines match up. The pieces you colored should be together, as
displayed above. The solid black lines represent the sides of the triangle that aren’t torn,
and the dashed lines represent where the triangle was torn.
What do you notice about the angles of the triangle when laid next to each other?
Together they create a straight line.
Use a protractor or a straight-edge to see that it creates a line of one hundred eighty
degrees.
What conclusion can we draw about the sum of the interior angles of a triangle? The sum
of the interior, or inside, angles of a triangle is always one hundred eighty degrees.
IDENTIFYING MISSING ANGLES IN TRIANGLES
What do you remember about the sum of the angles in any triangle? The sum of the
angles in any triangle is one hundred eighty degrees.
How can this information help us to find a missing angle in a triangle?
What is one thing you need to know in order to find a missing measure of a triangle?
You need to know two of the angles.
What else do you need to know in order to find a missing measure of a triangle? You
need to know that every triangle has a sum of one hundred eighty degrees.
Take a moment to write a plan for finding the missing angle in a triangle. What plan did
you write? When you know two angles of a triangle, subtract the sum from one hundred
eighty degrees to get the measure of the third angle.
What are the two known angles in this triangle? Seventy five degrees and thirty two
degrees
What needs to be done with the two known angles? Add them and then subtract the sum
from one hundred eighty degrees.
What is the sum of the two known angles? One hundred seven degrees
What is the measure of the missing angle? Seventy three degrees
What are the two known angles in this triangle? Fifty one degrees and ninety degrees.
We know that the second angle that is given is ninety degrees because it is marked in the
corner of the triangle with a small square.
What needs to be done with the two known angles? Add them and then subtract the sum
from one hundred eighty degrees.
What is the sum of the two known angles? One hundred forty one degrees
What is the measure of the missing angle? Thirty nine degrees
WRITING EQUATIONS TO FIND A MISSING ANGLE
We can use angle relationships such as those from the Warm-up and the number of
degrees in a triangle to write equations to find a missing angle in geometric figures.
What is the sum of any three angles in a triangle? Three angles equal a total of one
hundred eighty degrees in a triangle.
Describe the relationship between Angle a and Angle b in the drawing. Angles a and b
are supplementary. Explain your answer. Angle a and Angle b form a straight line and
have a sum of one hundred eighty degrees.
How can you use the relationships we know from the drawing to write equations to find
the measurements of Angles a and b?
How can I write an equation to find missing Angle a in the triangle? Eighty plus forty
plus a has to equal one hundred eighty.
Why is this equation correct? There are one hundred eighty degrees total, and all three
angles added together are equal to one hundred eighty.
What is my first step in solving the equation? Add the two constants to get: one hundred
twenty plus a equals one hundred eighty.
What is the next step in solving the equation? Subtract one hundred twenty from both
sides to get the value of a, which is equal to sixty degrees.
How can we use the information about the measure of Angle a to find the measure of
Angle b? We can write an equation to find the measure of Angle b.
Write an equation to find the measure of Angle b. Angle a plus Angle b equals one
hundred eighty. Why is this equation correct? Angle a and Angle b are supplementary,
which means together they add up to one hundred eighty degrees.
What is my first step in solving the equation? Substitute the value of sixty for a, which
gives us sixty plus b is equal to one hundred eighty.
What is the next step in solving the equation? Subtract sixty from both sides to get the
value of b, which is one hundred twenty degrees.
What is one angle relationship you see in the drawing? The three angles in the triangle
add to a sum of one hundred eighty degrees.
How can we write that information to find the measure of Angle b? We can write an
equation and solve for b.
Which angle measurement do I have to find first and why? Angle b because it is part of
the triangle, and I know the other two angles of the triangle.
How can I write an equation to find missing Angle b in the triangle? Seventy plus
seventy plus the measure of Angle b is equal to one hundred eighty.
Why is this equation correct? There are one hundred eighty degrees total, and all three
angles added together are equal to one hundred eighty.
What is the first step in solving the equation? Add the two constants to get one hundred
forty plus the measure of angle b is equal to one hundred eighty.
What is the next step in solving the equation? Subtract one hundred forty from both sides
to get: b is equal to forty degrees.
What is the relationship between Angle a and Angle b? Angle a and Angle b are vertical
angles.
What does this mean? Angle a and Angle b are opposite each other when two lines
intersect and so they are congruent.
If Angle a and Angle b are congruent, and we know that Angle b is equal to forty
degrees, what is the measure of Angle a? Angle a is also equal to forty degrees.
How could we use the information we know about Angle a and Angle b to find the
measure of Angle c? Angle b and Angle c are supplementary angles.
What does this mean? This means that they have a sum of one hundred eighty degrees.
How can we use this information to find the measure of Angle c? We can write an
equation and solve for c. Forty plus the measure of Angle c is equal to one hundred
eighty. The measure of Angle c is equal to one hundred forty degrees.
What is the relationship between Angle c and Angle d? They are vertical angles.
What is true about the measure of vertical angles? They are congruent.
What is the measure of Angle d? One hundred forty degrees.
SOLVE PROBLEM – COMPLETION
We are now going to go back to the SOLVE problem from the beginning of the lesson.
The question was, Samantha is designing a triangular garden bed in a new city park. The
drawing below shows where the garden bed meets a walking path. What is the
measurement of Angle c?
In the S Step, we Study the Problem. We underline the question and then we complete
this statement, this problem is asking me to find the angle measurement of Angle c.
In the O Step, Organize the Facts. Identify the facts. We go back to the original problem
and place a vertical line or strike mark at the end of each fact. Samantha is designing a
triangular garden bed in a new city park./ the drawing below shows where the garden
bed meets a walking path./
After we identify the facts we eliminate the unnecessary facts. We go back to the
original problem again and we read the facts. Samantha is designing a triangular garden
bed in a new city park. This fact is unnecessary, because we do not need that information
in order to determine the measurement of Angle c.
After we eliminate the unnecessary facts we list the necessary facts. We know that there
are three angles that make one hundred eighty degree total. They are fifty five degrees,
forty two degrees and Angle a. We also know that there are three angles of a triangle that
make one hundred eighty degrees: Angle a, Angle c and twenty seven degrees.
L, Line Up a Plan. Write in words what your plan of action will be. Set up an equation
to find the number of degrees in Angle a. Use that answer to set up an equation to find
the number of degrees in Angle c.
Choose an operation or operations. We’ll be using addition and subtraction.
V, Verify Your Plan with Action. Estimate your answer. Our estimate here is that Angle
c is acute about sixty degrees.
Carry out your plan. Angle a plus fifty five plus forty two is equal to one hundred eighty.
We combine our constants and solve for the value of a, which is eighty three degrees.
We know that Angle c plus a plus twenty seven is equal to one hundred eighty. We can
substitute in the value of eighty three for Angle a, to determine our value of c, which is
seventy degrees.
E, Examine Your Results.
Does your answer make sense? Compare your answer to the question. Yes, I found the
number of degrees for Angle c.
Is your answer reasonable? Compare your answer to the estimate. Yes, because seventy
degrees is close to sixty degrees.
Is your answer accurate? Check your work. Yes.
Write your answer in a complete sentence. Angle c has a measure of seventy degrees.
CLOSURE
Now let’s go back and discuss the essential questions from this lesson.
Our first question was, explain the meaning of complementary and supplementary angles.
Complementary angles are two angles that have a sum of ninety degrees. Supplementary
angles have a sum of one hundred eighty degrees.
Number two, explain the meaning of adjacent and vertical angles. Adjacent angles are
angles that share one ray. Vertical angles are angles that share a vertex, are opposite
from each other, and are congruent.
And number three, explain how you can use angle relationships and triangle relationships
to find the missing angle in a triangle. You can write an equation using what you know
about the measures of the angles in the relationship.