Download ANGLE RELATIONSHIPS – PART II INTRODUCTION The objective

ANGLE RELATIONSHIPS – PART II
INTRODUCTION
The objective for this lesson on Angle Relationships – Part II is, the student will explore angle relationships created by
parallel lines cut by a transversal in order to solve real world problems.
The skills students should have in order to help them in this lesson include Congruency, Vertical Angles, Supplementary
Angles and Adjacent Angles.
We will have three essential questions that will be guiding our lesson. Number one, what types of angle relationships
are created when we use a transversal to cut through parallel lines? Number two, if parallel lines are cut by only one
transversal, how many different measures of angles are created? And number three, where can parallel lines cut by a
transversal be seen in real life situations?
We will begin by completing the warm-up of using knowledge of angles, to prepare for angle relationships – part II in
this lesson.
SOLVE PROBLEM – INTRODUCTION
The SOLVE problem for this lesson is, a local park recently built a bridge over a small pond. They would like to provide a
railing for safety for visitors who choose to cross the bridge. The architecture calls for two parallel beams with a cross
beam. As the carpenters are constructing the bridge, they want to be sure that a consistent pattern is achieved with the
railing. If the measure of Angle B is forty one degrees, what should be the measure of Angle A? The following picture is
also provided with this problem.
We will begin by Studying the Problem. First we need to identify where the question is located within the problem, and
we will underline the question. The question for this problem is, what should be the measure of Angle A?
Now that we have identified where the question is located within the problem, we need to put this question in our own
words in the form of a statement. This problem is asking me to find the measure of Angle A.
During this lesson we will learn how to explore the relationships between angles that are formed with parallel lines and
transversals. We will use this knowledge to complete this SOLVE problem at the end of the lesson.
DISCOVERY ACTIVITY – ANGLE RELATIONSHIPS
On the blank page below the SOLVE problem, place your ruler horizontally as you see here. While holding the ruler in
place, use your pencil and the ruler to draw a line above the ruler and below the ruler. You should have two straight
lines that look like this. Now add directional arrows to the lines. What do these arrows mean? They mean that the line
continues on forever. Label the top line “a” and the bottom line “b.” Describe what we have created using the ruler as
our guide. We created a set of parallel lines.
Where might we see a real world example of parallel lines? Some examples may be, railroad tracks, the top and bottom
of a math book, the left and right edge of a desk, and many other examples.
Define parallel lines based on the real world examples you discussed. Parallel lines are a pair of lines that never
intersect.
We can use a symbol to identify a pair of lines as parallel by drawing a key on our diagram. To show that line a, is
parallel to line b, we can use a symbol showing two vertical lines between the letters representing each of the lines.
What we see here, is that line a, is parallel to line b, by using this symbol of the two vertical lines between these two
letters representing the lines. Let’s place this key on our diagram to show that line a, is parallel to line b.
Now, place the ruler so that it rises from left to right and is intersecting both of the parallel lines on the paper, as you
see here. Draw the intersecting line. Label this line “t.” Be sure to add arrows at the end of the line.
How can we describe the line we just drew? It intersects or crosses the two parallel lines. Why do you think we used
the letter “t” to represent the line that intersects the parallel lines? Think about this. Another word we could use for a
line that crosses a set of parallel lines is a transversal, which means a line that passes through two lines in the same
plane.
Let’s label each of the angles on the drawing using numbers so we can all refer to each angle. There are a total of eight
angles in our drawing.
Now using a protractor, measure angle one. Then record the measure in the section with Angle one. Let’s do this now.
Place your protractor so that you can measure Angle one. We see that the measure of Angle one in this drawing is one
hundred twenty nine degrees. Let’s label Angle one as equal to one hundred twenty nine degrees. What do you notice
about Angle one and Angle two? They form a straight line. What is the sum of the degrees of Angles one and two?
Justify your answer. The sum of Angles one and two is one hundred eighty degrees, because they create a straight line.
And we know that a straight line measures one hundred eighty degrees. If the total of the two angles is one hundred
eighty degrees, how can we find the measure of Angle two? We can subtract my measure for Angle one from one
hundred eighty degrees. So let’s now subtract the measure for Angle one from one hundred eighty degrees. One
hundred eighty minus one hundred twenty nine equals fifty one. The measure of Angle two is fifty one degrees. Let’s
label Angle two as equal to fifty one degrees.
Now place a sticky note over the top of Angle one. Using a pencil, trace Angle one on the sticky note. Use a ruler or
protractor as a straight edge for tracing. Label Angle one on the sticky note. Now let’s move Angle one’s sticky note off
to the side. Using the second sticky note, repeat the tracing process for Angle two. We will place the sticky note over
the top of Angle two and use a pencil to trace Angle two on to the sticky note. Be sure to label this angle as Angle two
on the sticky note. Now move your sticky note for Angle two off to the side.
Using the two angles that your traced, explore the measures of Angles three and four. An example of how you could
explore the relationship between the angles is seen here with the sticky note for Angle two. Do you notice any
connections with Angle three, using the sticky note for Angle two? Angle three is the same measure as Angle two. Let’s
label Angle three with the same measure as Angle two. Angle three is equal to fifty one degrees.
What relationship do Angles two and three share that allows them to have the same measure? They are vertical angles.
What is a description of vertical angles? They are angles opposite of each other when two lines intersect.
What do we know about vertical angles? We know that they are congruent.
What is another way we could have identified the measure of Angle three not knowing the measure of Angle two?
Angles one and three are supplementary because they create a straight line, therefore, we could again subtract the
measure of Angle one from one hundred eighty degrees to find the measure of Angle three.
Now let’s take a look at the relationship of Angle one to Angles three and/or Angle four. An example of the relationship
can be seen here. Do you notice any connections with Angle four using the sticky note for Angle one? Angle four is the
same measure as Angle one. Let’s label Angle four with the measure one hundred twenty nine degrees, since we know
that it has the same measure as Angle one. Explain what relationship Angles one and four share. They are vertical
angles. What relationship does Angle four share with Angle two and with Angle three? They are supplementary angles,
therefore each pair of angles should total one hundred eighty degrees.
Now take a look at the graphic organizer seen here. We will use this graphic organizer to help organize the information
that we are learning in this lesson. Here is the second part of the graphic organizer. What is the title of this graphic
organizer? It is titled Types of Angles.
In the top left box, what is the first type of angle? It is Vertical Angles. Describe vertical angles. These are angles that
are opposite of each other when two lines intersect. Name pairs of vertical angles. We have identified the following as
pairs of vertical angles from our drawing. Angle one and Angle four are vertical angles. And Angle two and Angle three
are vertical angles.
Can you identify any other vertical angles on our original drawing? Angle five and Angle eight are vertical angles. And
Angle six and Angle seven are vertical angles.
So let’s organize this information into our graphic organizer for vertical angles. A description of vertical angles is that
these are angles opposite of each other when two lines intersect. The pairs of vertical angles that we identified from the
drawing are that Angle one is congruent to Angle four. Angle two is congruent to Angle three. Angle five is congruent to
Angle eight and Angle six is congruent to Angle seven. Each of these pairs of angles is a pair of vertical angles.
Now explore Angles five through eight using your sticky notes. Here is one exploration that you can do with Angles five
through eight. What do you notice about Angle five? It is the same measure as Angle one. Let’s record the measure of
Angle five as one hundred twenty nine degrees. What do you notice about the sticky note if you started at Angle one
and then line it up with Angle five? You do not need to rotate it, only slide it into position.
What do you notice about Angle six? It is the same measure as Angle two. Let’s record Angle six as having a measure of
fifty one degrees. Just like the measure for Angle two. What do you notice about the sticky note if you start it at Angle
two and then line it up with Angle six? You do not need to rotate it, only slide it into position.
When angles are on the same side of the transversal with the same measure, they are called corresponding angles. How
can we describe corresponding angles? These are Angles on the same side of the transversal and the same placement in
relation to the parallel line.
What are some examples of congruent corresponding angles? Angle one and Angle five are congruent corresponding
angles. Angle two and Angle six are congruent corresponding angles. Angle three and Angle seven are congruent
corresponding angles. And Angle four and Angle eight are congruent corresponding angles.
Now let’s add the information that we know about corresponding angles to our graphic organizer. Corresponding angles
are angles on the same side of the transversal and same placement in relation to the parallel line.
The congruent corresponding angles that we have in our original drawing are, Angle one congruent to Angle five. Angle
two congruent to Angle six. Angle three congruent to Angle seven. And Angle four congruent to Angle eight.
How can we find the measure of Angle seven in our drawing? It is supplementary to Angle five and it is also the same as
Angle six because they are vertical angles. This means that the measure for Angle seven is fifty one degrees. Let’s label
the measure for Angle seven in our drawing. Angle seven is equal to fifty one degrees.
How can we find the measure of Angle eight? It is supplementary to both Angle seven and Angle six, and it is also the
same as Angle five because they are vertical angles. This means that the measure of Angle eight is equal to one hundred
twenty nine degrees, the same as the measure for Angle five. Let’s label the measure for Angle eight in our drawing.
Angle eight is equal to one hundred twenty nine degrees.
What do you notice about Angles four and five? They are congruent because they have the same measure. What do
you notice about Angles three and six? They are congruent because they have the same measure. These angles share a
relationship. What do you think we can call these angles that are on opposite sides of the transversal and inside of the
parallel lines? These are called alternate interior angles. How can we describe these angles? These are angles on
opposite sides of the transversal inside of the parallel lines.
What are the congruent alternate interior angles? Angle three and Angle six are congruent alternate interior angles.
And Angle four and Angle five are congruent alternate interior angles.
Let’s include this information in our graphic organizer for alternate interior angles. A description of these angles is that
they are angles on opposite sides of the transversal inside of the parallel lines. The congruent angles that we found in
our drawing that are alternate interior angles are Angle three congruent to Angle six. And Angle four congruent to Angle
five.
Now looking back at our drawing again, what do you notice about Angles one and eight? They are congruent because
they have the same measure.
What do you notice about Angles two and seven? They are congruent because they have the same measure. These
angles share a relationship. What do you think we can call these angles that are on opposite sides of the transversal and
outside of the parallel lines? They are called alternate exterior angles. They are on the exterior side of the parallel lines.
How can we describe these angles? These are angles on opposite sides of the transversal outside of the parallel lines.
What are the congruent alternate exterior angles? Angle one and Angle eight are congruent alternate exterior angles.
And Angle two and Angle seven are congruent alternate exterior angles.
Let’s add this information to our graphic organizer as well for alternate exterior angles. A description of alternate
exterior angles is that these are angles on opposite sides of the transversal outside of the parallel lines. The examples of
alternate exterior angles in our drawing are, Angle one congruent to Angle eight, and Angle two congruent to Angle
seven.
So what do you notice about the number of different measures of angles on the diagram we created? There are only
two different measures. Some of the angles measure one hundred twenty nine degrees, and some of the angles
measure fifty one degrees. Therefore we can use all of our different angle relationships to find the measures of the
angles.
HORIZONTAL PARALLEL LINES – FINDING MISSING ANGLES
Take a look at the drawing seen here for Question one. What does the notation in the bottom right corner of the box
for Question one represent? It tells us that line m and line n are parallel to each other.
What is the angle measure we are given? We are told that angle three is equal to forty three degrees. Let’s record this
information in the box provided. Angle three equals forty three degrees.
Now let’s see if we can use the information we know about angle relationships to find the measure of the other angles
in the drawing.
What relationship do Angle one and Angle three share? They are supplementary. This means that together Angle one
and Angle three equal one hundred eighty degrees. So how can we find the measure of Angle one? We can subtract
one hundred eighty minus forty three which equals one hundred thirty seven degrees. The measure of Angle one is
equal to one hundred thirty seven degrees. Record this information in the box provided.
Now what is the relationship between Angle one and Angle four, as well as Angle three and Angle two? They are vertical
angles which means, they are congruent. Angle three and Angle two are congruent to each other. So the measure of
Angle two is the same as the measure of Angle three. Angle two equals forty three degrees. The same is true for Angle
one and four as these are vertical angles and are congruent. Angle four will have the same measure as Angle one. Angle
four equals one hundred thirty seven degrees. Be sure to record this information in the box provided.
Now how does Angle three compare to Angle six? They are congruent because they are alternate interior angles. The
measure of Angle six will be the same as the measure of Angle three. Angle six equals forty three degrees.
What is another pair of alternate interior angles? Angle four and Angle five are alternate interior angles. What can we
record as the measure of Angle five? One hundred thirty seven degrees, because alternate interior angles are
congruent. Record the measure of Angle five in the box provided.
We have two angles left that we need to find the measurements for. How can we find the measures for Angle seven and
eight? We could use the idea of vertical angles from Angle five and six, or alternate exterior angles from Angles one and
two, or corresponding angles from Angles three and four. Let’s use vertical angles. Angle seven will have the same
measure as Angle six, because vertical angles are congruent. The measure of Angle seven is forty three degrees. For
Angle eight let’s use alternate exterior angles. The alternate exterior angle to Angle eight is Angle one. Since alternate
exterior angles are congruent we know that the measure of Angle eight will be the same as the measure of Angle one.
So the measure of Angle eight is one hundred thirty seven degrees.
We started with our drawing and the measurement for one of the angles. We used this measurement to find the
measure of the other seven angles in the drawing.
VERTICAL PARALLEL LINES – FINDING MISSING ANGLES
Take a look at the drawing provided here. Are lines j and k parallel? Yes they are. Explain and defend your thinking.
The two arrows at the top of the parallel lines are symbols that the lines are parallel.
How is this diagram different from the other sets of parallel lines that we have seen? In this case, the parallel lines are
vertical rather than horizontal.
Where are the exterior angles located? They are to the right of line k and to the left of line j.
Let’s use our knowledge of angle relationships to find the missing angles and then complete the tables. What angle
measure are we given? We are given that Angle seven is one hundred thirty three degrees. Record the measure of
Angle seven in the box provided. Angle seven equals one hundred thirty three degrees.
So what is the measure of Angle six? The measure of Angle six is forty seven degrees. Defend your answer. It is
supplementary with Angle seven, therefore we can subtract one hundred eighty minus one hundred thirty three to find
the degrees for Angle six. Record the measure of Angle six in the box provided. Angle six equals forty seven degrees.
So, how can we identify the measures for Angles five and eight? Explain your thinking. Angle five is the same as Angle
seven and Angle six is the same as Angle eight because they are vertical angles. Since Angle five is the same as Angle
seven the measure of Angle five is one hundred thirty three degrees. And since Angle eight is the same as Angle six the
measure of Angle eight is forty seven degrees. Record the measure of both of these angles in the box provided.
What are the other pairs of vertical angles in the drawing? Angle two and Angle four, as well as Angle one and Angle
three.
Now take a look at the graphic organizer. Let’s record the vertical angles from the drawing. Angle one is congruent to
Angle three. Angle two is congruent to Angle four. Angle five is congruent to Angle seven. And Angle six is congruent to
Angle eight. These are the vertical angles from the drawing.
Take a look at the drawing again. How does Angle three relate to Angle seven? Justify your answer. They are
corresponding angles so they are congruent. Since Angle three is congruent to Angle seven we know that the measure
of Angle three is one hundred thirty three degrees. Record the measure of Angle three in the box provided.
What are some other corresponding angles? Angle six and Angle two are corresponding angles. Angles eight and Angle
four are corresponding angles. And Angle five and Angle one are corresponding angles. Let’s include this information
for Corresponding Angles in the graphic organizer. The corresponding angles are, Angle one congruent to
Angle five. Angle two congruent to Angle six. Angle three congruent to Angle seven. And Angle four congruent to Angle
eight.
Using corresponding angles, we are able to identify all of the missing measures in the box. Since Angle one is congruent
to Angle five, we know that the measure of Angle one will be one hundred thirty three degrees. Angle two is congruent
to Angle six so the measure of Angle two is forty seven degrees. And Angle four is congruent to Angle eight so we know
that the measure of Angle four is forty seven degrees. You should now have recorded the measure of all of the angles in
the box provided.
Now let’s talk about the other angle relationships that we have in our drawing. We also have alternate interior angles in
our diagram. What angles are these? Angle three and Angle five are alternate interior angles. And Angle four and Angle
six are alternate interior angles. Let’s include this information in the graphic organizer. The alternate interior angles are
Angle three congruent to Angle five and Angle four congruent to Angle six.
The last group is alternate exterior angles. What angles fall into this category? Angle one and Angle seven are alternate
exterior angles. And Angle two and Angle eight are alternate exterior angles. We can finish our graphic organizer by
including these angles as alternate exterior angles. Angle one is congruent to Angle seven and Angle two is congruent to
Angle eight.
MULTIPLE TRANSVERSALS PROBLEM SOLVING
Take a look at the drawing seen here. We are going to use angle relationships from parallel lines cut by a transversal
and the angles above to prove that Angle one plus Angle two plus Angle three is equal to one hundred eighty degrees.
Are the parallel lines horizontal or vertical? They are horizontal. Line p is parallel to line q.
What is different about this diagram from the others that we looked at in this lesson? There are two lines that intersect
the parallel lines and they form a triangle.
Are we given any of the measures of any angles? No. What do we know about Angle three, Angle four and Angle five?
Defend your answer. They form a straight line, which means that their sum is one hundred eighty degrees.
What are we trying to prove in this example? We are trying to prove that Angles one, two and three also have a sum of
one hundred eighty degrees.
Can we identify any type of relationship between the sum of Angles three, four and five, and the sum of Angles one, two
and three? Yes, we can use angle relationships from parallel lines cut by a transversal. Explain how we can informally
prove the statement correct. If we can prove that Angles one, two and three are equal to Angles three, four and five,
then we can prove the statement correct.
What angle relationships can we see from the diagram? Angle one and Angle five are corresponding angles, which
means that they are congruent.
Are there any other angle relationships we can identify? Angle two and Angle four are alternate interior angles, which
means that they are congruent.
So what do these relationships tell us? Justify your answer. If Angle one is congruent to Angle five and Angle two, is
congruent to Angle four, then Angle one plus Angle two plus Angle three is equal to Angle five plus Angle four plus Angle
three which we know is equal to one hundred eighty degrees.
Can you identify another angle relationship with triangles that would verify this informal proof? The sum of interior
angles of any triangle is always one hundred eighty degrees.
PROBLEMS WITH PARALLEL LINES CUT BY TRANSVERSALS
Let’s read the problem together. Designers of upholstery are adding decorative stripes to the back of a chair. Two
parallel stripes are added. What should Angle B measure based on the angle given and the fact that the legs of the chair
stand parallel?
What is the problem asking us to find? It’s asking us to find the measure of Angle B.
What do we know about the stripes? Defend your answer. The two stripes are parallel lines. We can mark the two
stripes as parallel lines by placing an arrow on each of these lines.
What do we know about the legs of the chair? They are vertical parallel lines. Let’s include a symbol to show that these
are parallel lines by placing arrows on each of these lines as well.
What information are we given? We have one of the angles measuring sixty three degrees.
So if we were to use the legs as the parallel lines and the tip stripe as the transversal, what angle can we identify? We
can identify the alternate interior angle from the top stripe which will also measure sixty three degrees. Let’s mark this
angle as sixty three degrees, because alternate interior angles are congruent.
Knowing this angle, how can we find Angle B? Explain your thinking. The new angle we just found is a corresponding
angle to Angle B, therefore, they are congruent.
So what is the measure of Angle B? It is also sixty three degrees.
So looking at this problem our solution is, the alternate interior angle to sixty three degrees can be labeled on the top
stripe. We know that the stripes are also parallel, so we can identify Angle B as a corresponding angle to the alternate
interior angle. Therefore, Angle B is also sixty three degrees.
SOLVE PROBLEM – COMPLETION
We are now going to go back to the SOLVE problem from the beginning of the lesson. A local park recently built a bridge
over a small pond. They would like to provide a railing for safety for visitors who choose to cross the bridge. They
architecture calls for two parallel beams with a cross beam. As the carpenters are constructing the bridge, they want to
be sure that a consistent pattern is achieved with the railing. If the measure of Angle B is forty one degrees, what should
be the measure of Angle A?
At the beginning of the lesson we Study the Problem. First we identified where the question was located within the
problem and we underlined the question. We then put this question in our own words in the form of a statement. This
problem is asking me to find the measure of Angle A. The diagram seen here was also provided with the problem.
Now in Step O, we will Organize the Facts. First we need to identify the facts. A local park recently built a bridge over a
small pond/ fact. They would like to provide a railing for safety for visitors who choose to cross the bridge/ fact. The
architecture calls for two parallel beams with a cross beam/fact. As the carpenters are constructing the bridge, they
want to be sure that a consistent pattern is achieved with the railing/fact. If the measure of Angle B is forty one
degrees/fact, what should be the measure of Angle A?
Now that we have identified the facts, we are ready to eliminate the unnecessary facts. These are the facts that will not
help us to find the measure of Angle A. A local park recently built a bridge over a small pond. Will this fact help us to
find the measure of Angle A? No, so we will eliminate the fact. They would like to provide a railing for safety for visitors
who choose to cross the bridge. Knowing that the railing is being provided will not help is to find the measure for Angle
A. So we will eliminate this fact. The architecture calls for two parallel beams with a cross beam. Will this information
help us to find the measure of Angle A? Knowing that the beams are parallel to each other will help us to find the
measure of Angle A, so we will keep this fact. As the carpenters are constructing the bridge, they want to be sure that a
consistent pattern is achieved with the railing. This information is not going to help us to find the measure of Angle A, so
we will eliminate this fact. If the measure of Angle B is forty one degrees, knowing the measure of Angle B will help us
to find the measure of Angle A. So we will keep this fact.
Now that we have eliminated the unnecessary facts, we are ready to list the necessary facts. These facts are that the
beams are parallel and that Angle B is equal to forty one degrees.
Now in Step L, we need to Line Up a Plan. First we will write in words what your plan of action will be. We need to
identify an angle relationship between Angles A and B to identify the measure of Angle A. What operation or operations
will be used in this plan? There is not a specific operation or operations. We will use the diagram that was provided to
help us to solve the problem.
In Step V, we Verify Your Plan with Action. First estimate your answer. If you look at the diagram it looks like Angle A
and Angle B are the same. So we can estimate that Angle A is equal to Angle B.
Now let’s carry out your plan. In our plan we said that we needed to identify an angle relationship between Angles A
and B, to identify the measure of Angle A. Angle B and Angle A are alternate interior angles, which means they are
congruent. Therefore, Angle A is also forty one degrees.
Now in Step E, we Examine Your Results.
Does your answer make sense? Here compare your answer to the question. Yes, because we found the measure of
Angle A.
Is your answer reasonable? Here compare your answer to the estimate. Yes, because it matches my estimate that
Angle A and Angle B are equal.
And is your answer accurate? Yes
We are now ready to write your answer in a complete sentence. Angle A should be forty one degrees.
CLOSURE
Now let’s go back and discuss the essential questions from this lesson.
Our first question was, what types of angle relationships are created when we use a transversal to cut through parallel
lines? The transversal will create vertical angles, corresponding angles, alternate interior angles, alternate exterior
angles and supplementary angles.
Our second question was, if parallel lines are cut by only one transversal, how many different measures of angles are
created? Only two different measures will be created from the transversal. We can use angle relationships to identify
the measurements.
And our third question was, where can parallel lines cut by a transversal be seen in real life situations? Parallel lines
with transversals can be seen with bracing on architecture, fences or railings, maps with streets and buildings, and many
other places.