TOPIC 10
Parallel lines
What are parallel lines? Write the definition:
You will need to look up the defintions of these pairs of angles. See Topic 1 (pages 9 and 10)
Name pairs of corresponding angles:
Name pairs of alternate interior angles:
Name pairs of alternate exterior angles:
Name pairs of same side interior angles:
Name pairs of same side exterior angles:
Topic 10 (Parallel Lines)
page 2
We will name pairs of angles again, ut this time it is announced which pairs of lines (and a transversal)
are he ones to which we are applying the names.
Name pairs of corresponding angles with respect to (wrt) a and b:
Name pairs of corresponding angles with respect to (wrt) c and d:
Name pairs of alternate interior angles with respect to (wrt) a and b:
Name pairs of alternate interior angles with respect to (wrt) c and d:
Name pairs of alternate exterior angles with respect to (wrt) c and d:
Name pairs of same side interior angles with respect to (wrt) a and b:
Name pairs of same side exterior angles with respect to (wrt) c and d:
Topic 10 (Parallel Lines)
page 3
Investigation with Parallel Lines and Associated Angles
SKETCHPAD
1.
Using Sketchpad, draw two lines which are not parallel, and draw a third line
which crosses these two. (The third line is called a transversal.)
2.
Construct enough points (8) to label all of the angles shown. (Note that your
angles will not be called < 1,<2, <3, etc..)
(Note: Use the point tool to place a point on the lines. Use the text tool to label
the points. If the points are not labeled as above, you can double click on the
label and a box will pop up and you can rename the points.)
3.
Record 3 sets of measurements. Measure all 8 angles, and record your results below.
(Record a set, move one of the lines, record measurements again, move one line, and
record the measurements again.
m< 1 =
m< 2 =
m< 3 =
m< 4 =
m< 5 =
m< 6 =
m< 7 =
m< 8 =
m< 1 =
m< 2 =
m< 3 =
m< 4 =
m< 5 =
m< 6 =
m< 7 =
m< 8 =
m< 1 =
m< 2 =
m< 3 =
m< 4 =
m< 5 =
m< 6 =
m< 7 =
m< 8 =
5.
Save your Sketchpad document as NONPARALLEL. I may ask to see your
document in class.
6.
Using Sketchpad, construct ABD first. Mark points A, B, and D on the line.
Choose a point H which is not on AB , using the point tool. Construct segment
Topic 10 (Parallel Lines)
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CBHF (Highlight A and C, and construct a line.). By the way, I am following
the drawing above, except this time there will be a pair of parallel lines. Choose H
and ABD , and construct a line through H parallel to ABD . Construct other points in the
drawing above. (You can extend the line segments, or you can make them into lines, if
necessary, in order to name all angles with three letters, as Sketchpad requires you to do. You
cannot use the numbered names for angles in Sketchpad.
7.
Measure all of the angles three times, as you did in the investigation above.
m< 1 =
m< 2 =
m< 3 =
m< 4 =
m< 5 =
m< 6 =
m< 7 =
m< 8 =
m< 1 =
m< 2 =
m< 3 =
m< 4 =
m< 5 =
m< 6 =
m< 7 =
m< 8 =
m< 1 =
m< 2 =
m< 3 =
m< 4 =
m< 5 =
m< 6 =
m< 7 =
m< 8 =
8.
Save your Sketchpad document as PARALLELS.
9.
QUESTIONS:
10.
a)
Are there any congruent angles? If so, which ones?
b)
Are there any supplementary angles? If so, which ones?
c)
Are there any complementary angles? If so, which ones?
Look at your two investigations above----one with non parallel lines cut by a
transversal, and the other with parallel lines cut by a transversal.
See if there are some conclusions that appear to be true (conjectures).
Angles
With non-parallel lines
With parallel Lines
___________________________________________________________
corresponding
alternate interior
alternate exterior
same-side interior
Topic 10 (Parallel Lines)
page 5
same-side exterior
11.
State these conjectures in the IF … THEN … format
If lines are parallel and cut by a transversal, then corresponding angles are
congruent.
Add other similar statements……
If lines are parallel and cut by a transversal, then alternate interior angles are congruent.
If lines are parallel and cut by a transversal, then alternate exterior angles are congruent.
If lines are parallel and cut by a transversal, then same side interior angles are supp.
If lines are parallel and cut by a transversal, then same side exterior angles are supp.
Taking all of these conjectures that you have just made, we can accept ONE of them as a postulate and
prove the others. So there will be one postulate and several theorems. Which one is accepted as the
postulate is arbitrary. Modern textbooks and traiditon all accept one particular one, and the others are
proved as theorems. In Euclid’s work The Elements, it is adifferent one that is listd as the postulate.
His is called the Parallel Postulate, and it is his famout Postulate #5. (We will discuss that postulate
and its ramificaions later.)
We will add the postulate and theorems now:
POSTULATE: If lines are parallel and cut by a transversal, then corresponding angles are
congruent.
THEOREM: If lines are parallel and cut by a transversal, then alternate interior angles are
congruent.
THEOREM: If lines are parallel and cut by a transversal, then alternate exterior angles are
congruent.
THEOREM: If lines are parallel and cut by a transversal, then same side interior angles are supp.
Topic 10 (Parallel Lines)
page 6
THEOREM: If lines are parallel and cut by a transversal, then same side exterior angles are supp.
We will prove each of the theorems, so that they can be added to our repertoire. You will need to write
your proofs in your notebook. I have not provided adequate space for the proofs here.
Then, you will be provided with several proofs involving parallel lines. You will be paired with a
classmate each day in class. You will complete a few proofs, and your partner will do the same with
different proofs. Then you will critique hers, and she will critique yours. The “critiquer” is looking for
accuracy of the logic, the notation, and the success of the proof. After both of you have finished work
on a proof, you will hand it in to me to evaluate. You are evaluated both on your proof and your
critique of another proof. When we are finished, you will have seen nearly all of the proofs you need.
Looking back at coordinate geometry, what is true about lines which are parallel?
Topic 10 (Parallel Lines)
Why is this result true? Let’s look at an explanation (“ a proof”).
page 7
Topic 10 (Parallel Lines)
page 8
So, if lines are parallel, then coresponding angles are congruent ( P  CAC). If corresponding angles
are congruent, then the slopes of the lines are equal. By a syllogism, if lines are parallel, then slopes
are equal.
Parallel Lines and Congruent Angles
(Numerical Problems)
Find the value of x and y.
12.
13.
14.
15.
16.
17.
Topic 10 (Parallel Lines)
page 9
Find the measure of each numbered angle. You can solve the problem on the drawing, if that works
for you, but list the answers off to the side.
18.
19.
20.
21.
22.
23.
Topic 10 (Parallel Lines)
page 10
Converses of the postulate and theorems: Are the converses of the statements above
TRUE or FALSE?
Let’s start with an investigation. Use a straightedge and a protractor. Use the given line, a, and the
given transversal, t. If you draw a line, m, with a vertex at B so that the corresponding angles are
congruent, does it appear that the line m is parallel to a?
If it does appear reasonable that a || b, then we will state the following postulate:
If corresponding angles are congruent, then lines are parallel.
(Note: This requires that there be two lines cut by a transvesal that create the corresponding angles
which are congriuent. We could include all of this details in the statement of the postulate, but we
sometimes realize that there would not even be angles if there were not a transveral. So we sometimes
do not always state that “a transversal cuts the two lines”, even though there has to be a transveral.)
24.
Now, write the converse of all of the theorems above. Your job will be to decide and prove
which of them are true in this converse form.
Topic 10 (Parallel Lines)
page 11
Finally, let’s look at the properties called reflexive, symmetric, and transitive for parallelism.
25.
Is parallelism reflexive?
if you think it is false.
Explain why it is true, or provide a counterexample or explanation
26.
Is parallelism symmetric?
if you think it is false.
27.
Is parallelism transitive?
if you think it is false.
28.
For the last one, prove that it is true. (OK, I gave away the correct conclusion!!)
Explain why it is true, or provide a counterexample or explanation
Explain why it is true, or provide a counterexample or explanation
Topic 10 (Parallel Lines)
page 12
More problems with parallel lines and associated angles.
A
F
13
12
9
11 10
14 B
8
2
1
E
29.
30.
7
3
D
4 5
G
6
C
True or False for each conclusion
Given: AB || EC
a) 12  3
b) 1   4
c) 9  6
d) 13 and 8 are supplementary
e) 13  1
Which one pair of lines would need to be parallel to make each conclusion true?
a) 11  2
b) 10  7
c) 9  6
d) BFG  FGD
e) 1 and 5 are supplementary
f) 1  BCG
g) 14  13
Find the measure of RST in the figure below.
31.
R
120
S
160
T
32. Given: l || AB
Find the values of x and y.
l
30
5y
2x
(x – y )
B
A
A
33. Given: AB || DF
R
1
C
2
RC || ED
Prove: m1  m4
B
E
3
4
F
D
Topic 10: Parallel Lines
34.
35.
page 14
Construct two parallel lines cut by a transversal. Next construct the angle bisectors of any
pair of corresponding angles. What appears to be true about the bisectors?
Prove that your conclusion is true.
Topic 10: Parallel Lines
page 15
36.
Construct two parallel lines cut by a transversal. Next construct the angle bisectors of any
pair of alternate interior angles. What appears to be true about the bisectors?
37.
Prove that your conclusion is true.
Topic 10: Parallel Lines
page 16
Finding the Circumference of the Earth
Eratosthenes used the ideas about corresponding angles and perhaps alternate interior
angles to calculate the circumference of the earth 2200 years ago. Imagine the technology
that he used… or that he did not have to use at the time.
The accomplishment that Eratosthenes is probably most
well known for was his estimation of the circumference of
the Earth. Although not many believed it at the time,
Eratosthenes knew that the world was round which allowed
him to make this discovery. He knew from observation that
in the town of Syene in Egypt, the sun shone straight down
the wells because of the way the sun reflected. He realized
that because of the huge distance between the sun and the
earth, all light waves must be parallel. Because the earth
was curved, the light would hit a different city at a different
angle. Eratosthenes went to Alexandria and found out that
the light hit the earth at an angle of 7.2 degrees. This
meant that if he continued the light waves at Alexandria
and the light waves at Syene, they would meet at a 7.2
degree angle at the center of the earth. Since 7.2 degrees is one fiftieth of a circle (360 degrees), the
distance between Alexandria and Syene would be one fiftieth of the circumference of the earth. He
measured this distance as approximately 5000 stadia, a measurement equal to about 500 miles or 800
kilometers. When he multiplied this distance, he got about 250,000 stadia, equal to 25,000 miles or
40,000 kilometers. The actual circumference around the equator is about 24,900 miles, which means that
Eratosthenes was incredibly accurate in his estimation.
Note: There is much debate among scientists and historians of today about the length of a stadia,
because with some proposed lengths Eratosthenes would have been off in his measurement by more
than 100 miles. However, in any case it is a remarkable estimate for someone to have made during the
time period.
From http://eratosthenes.weebly.com/accomplishments.html
38.
Do a search to see if you can find out where Alexandria, Egypt and Syene, Egypt are.
(Syene is near the modern day Aswan, Egypt.) How far apart are they?
Topic 10: Parallel Lines
page 17
From http://www.3villagecsd.k12.ny.us/wmhs/Departments/Math/OBrien/eros.html
Questions:
39.
What assumption(s) are made about the route from Syene to Alexandria?
40.
How were measurements of angles and distances made at the time that Eratosthenes did
them?
41.
Does it appear apparent that Eratosthenes knew about angles associated with parallel
lines?
42.
How readily do you think that his results were accepted in his time?