Looking More Carefully at Parallel Lines
Exchange Problem Sets from Lesson 17 with a classmate. Solve the problems posed by your classmate while he or she solves yours. Compare your solutions, and then discuss and resolve any discrepancies. Why were you asked only to locate the point of rotation rather than to rotate a pre-image to obtain the image? How did you use perpendicular bisectors in constructing your solutions?
We say that two lines are parallel if they lie in the same plane and do not intersect. Two segments or rays are parallel if the lines containing them are parallel.
Why is the phrase in the plane critical to the definition of parallel lines? Explain and illustrate your reasoning.
In Lesson 7, we recalled some basic facts learned in earlier grades about pairs of lines and angles created by a transversal to those lines. One of those basic facts is the following:
Suppose a transversal intersects a pair of lines. The lines are parallel if and only if a pair of alternate interior angles are equal in measure.
Our goal in this lesson is to prove this theorem using basic rigid motions, geometry assumptions, and a geometry assumption we introduce in this lesson called the parallel postulate. Of all of the geometry assumptions we have given so far, the parallel postulate gets a special name because of the special role it played in the history of mathematics. (Euclid included a version of the parallel postulate in his books, and for 2,000 years people tried to show that it was not a necessary assumption. Not only did it turn out that the assumption was necessary for Euclidean geometry, but study of the parallel postulate led to the creation of non-Euclidean geometries.)
The basic fact above really has two parts, which we prove separately:
Given a line 𝑙 and a point 𝑃 not on the line, follow the steps below to rotate 𝑙 by 180° to a line 𝑙′
that passes through 𝑃:
a. Label any point 𝐴 on 𝑙.
b. Find the midpoint of segment 𝐴𝑃 using a ruler. (Measure the length of segment 𝐴𝑃, and locate the point that is distance 𝐴𝑃/2 from 𝐴 between 𝐴 and 𝑃.) Label the midpoint 𝐶.
c. Perform a 180° rotation around center 𝐶. To quickly find the image of 𝑙 under this rotation by hand:
i. Pick another point 𝐵 on 𝑙.
ii. Draw 𝐶𝐵.
iii. Draw circle: center 𝐶, radius 𝐶𝐵.
iv. Label the other point where the circle intersects 𝐶𝐵 by 𝑄.
v. Draw 𝑃𝑄.
d. Label the image of the rotation by 180° of 𝑙 by 𝑙′ = 𝑅𝐶,180(𝑙).
How does your construction relate to the geometry assumption stated above to rotations? Complete the statement below to clarify your observations:
𝑅𝐶,180 is a 180° _____ around 𝐶. Rotations preserve ____ ; therefore 𝑅𝐶,180, maps the line 𝑙 to the line ____ . What is R𝐶,180(𝐴)?
The lines 𝑙 and 𝑙′ in the construction certainly look parallel, but we do not have to rely on looks.
Claim: In the construction, 𝑙 is parallel to 𝑙′.
PROOF: We show that assuming they are not parallel leads to a contradiction. If they are not parallel, then they must intersect somewhere. Call that point 𝑋. Since 𝑋 is on 𝑙′, it must be the image of some point 𝑆 on 𝑙 under the 𝑅𝐶,180 rotation, (i.e., 𝑅𝐶,180(𝑆) = 𝑋). Since 𝑅𝐶,180 is a 180° rotation, 𝑆 and 𝑋 must be the endpoints of a diameter of a circle that has center 𝐶. In particular, 𝑆𝑋 must contain 𝐶. Since 𝑆 is a point on 𝑙, and 𝑋 is a different point on 𝑙 (it was the intersection of both lines), we have that 𝑙 = 𝑆𝑋 because there is only one line through two points. But 𝑆𝑋 also contains 𝐶, which means that 𝑙 contains 𝐶. However, 𝐶 was constructed so that it was not on 𝑙. This is absurd.
There are only two possibilities for any two distinct lines 𝑙 and 𝑙′ in a plane: either the lines are parallel, or they are not parallel. Since assuming the lines were not parallel led to a false conclusion, the only possibility left is that 𝑙 and 𝑙′ were parallel to begin with.
The construction and claim together implies the following theorem.
THEOREM: Given a line 𝑙 and a point 𝑃 not on the line, then there exists line 𝑙′ that contains 𝑃 and is parallel to 𝑙.
This is a theorem we have justified before using compass and straightedge constructions, but now we see it follows directly from basic rigid motions and our geometry assumptions.
We are now ready to prove the first part of the basic fact above. We have two lines, 𝑙 and 𝑙′, and all we know is that a transversal 𝐴𝑃 intersects 𝑙 and 𝑙′ such that a pair of alternate interior angles are equal in measure. (In the picture below, we are assuming 𝑚∠𝑄𝑃𝐴 = 𝑚∠𝐵𝐴𝑃.)
Let 𝐶 be the midpoint of 𝐴𝑃. What happens if you rotate 180° around the center 𝐶? Is there enough information to
show that 𝑅𝐶,180(𝑙) = 𝑙′?
a. What is the image of the segment 𝐴𝑃?
b. In particular, what is the image of the point 𝐴?
c. Why are the points 𝑄 and 𝑅𝐶,180(𝐵) on the same side of 𝐴𝑃 ?
d. What is the image of 𝑅𝐶,180(∠𝐵𝐴𝑃)? Is it ∠𝑄𝑃𝐴? Explain why.
e. Why is 𝑅𝐶,180(𝑙) = 𝑙′?
We have just proven that a rotation by 180° takes 𝑙 to 𝑙′. By the claim in Example 3, lines 𝑙 and 𝑙′ must be parallel, which is summarized below.
THEOREM: Suppose a transversal intersects a pair of lines. If a pair of alternate interior angles are equal in measure, then the pair of lines are parallel.
In Example 5, suppose we had used a different rotation to construct a line parallel to 𝑙 that contains 𝑃. Such constructions are certainly plentiful. For example, for every other point 𝐷 on 𝑙, we can find the midpoint of segment 𝑃𝐷 and use the construction in Example 2 to construct a different 180° rotation around a different center such that the image of the line 𝑙 is a parallel line through the point 𝑃. Are any of these parallel lines through 𝑃 different? In other words,
Can we draw a line other than the line 𝑙′ through 𝑃 that never meets 𝑙?
The answer may surprise you; it stumped mathematicians and physicists for centuries. In nature, the answer is that it is sometimes possible and sometimes not. This is because there are places in the universe (near massive stars, for example) where the model geometry of space is not plane-like or flat but is actually quite curved. To rule out these other types of strange but beautiful geometries, we must assume that the answer to the previous question is only one line. That choice becomes one of our geometry assumptions:
(Parallel Postulate) Through a given external point there is at most one line parallel to a given line.
In other words, we assume that for any point 𝑃 in the plane not lying on a line ℓ, every line in the plane that contains 𝑃 intersects ℓ except at most one line—the one we call parallel to ℓ.
We can use the parallel postulate to prove the second part of the basic fact.
THEOREM: Suppose a transversal intersects a pair of lines. If the pair of lines are parallel, then the pair of alternate interior angles are equal in measure.
PROOF: Suppose that a transversal 𝐴𝑃 intersects line 𝑙 at 𝐴 and 𝑙′ at 𝑃, pick and label another point 𝐵 on 𝑙, and choose a point 𝑄 on 𝑙′ on the opposite side of 𝐴𝑃 as 𝐵. The picture might look like the figure below:
Let 𝐶 be the midpoint of 𝐴𝑃, and apply a rotation by 180° around the center 𝐶. As in previous discussions, the image of 𝑙 is the line 𝑅𝐶,180(𝑙), which is parallel to 𝑙 and contains point 𝑃. Since 𝑙 ′ and 𝑅𝐶,180(𝑙) are both parallel to 𝑙 and contain 𝑃, by the parallel postulate, they must be the same line: 𝑅𝐶,180(𝑙) = 𝑙 ′ . In particular, 𝑅𝐶,180(∠𝐵𝐴𝑃) = ∠𝑄𝑃𝐴. Since rotations preserve angle measures, 𝑚∠𝐵𝐴𝑃 = 𝑚∠𝑄𝑃𝐴, which was what we needed to show.
It is important to point out that, although we only proved the alternate interior angles theorem, the same sort of proofs can be done in the exact same way to prove the corresponding angles theorem and the interior angles theorem. Thus, all of the proofs we have done so far (in class and in the Problem Sets) that use these facts are really based, in part, on our assumptions about rigid motions.
We end this lesson with a theorem that we just state but can be easily proved using the parallel postulate. THEOREM: If three distinct lines 𝑙1 , 𝑙2 , and 𝑙3 in the plane have the property that 𝑙1 ∥ 𝑙2 and 𝑙2 ∥ 𝑙3 , then 𝑙1 ∥ 𝑙3 . (In proofs, this can be written as, “If two lines are parallel to the same line, then they are parallel to each other.”)
PARALLEL: Two lines are parallel if they lie in the same plane and do not intersect. Two segments or rays are parallel if the lines containing them are parallel lines.
TRANSVERSAL: Given a pair of lines 𝑙 and 𝑚 in a plane, a third line 𝑡 is a transversal if it intersects 𝑙 at a single point and intersects 𝑚 at a single but different point. The definition of transversal rules out the possibility that any two of the lines 𝑙, 𝑚, and 𝑡 are the same line.
ALTERNATE INTERIOR ANGLES: Let line 𝑡 be a transversal to lines 𝑙 and 𝑚 such that 𝑡 intersects 𝑙 at point 𝑃 and intersects 𝑚 at point 𝑄. Let 𝑅 be a point on 𝑙 and 𝑆 be a point on 𝑚 such that the points 𝑅 and 𝑆 lie in opposite half planes of 𝑡. Then the ∠𝑅𝑃𝑄 and the ∠𝑃𝑄𝑆 are called alternate interior angles of the transversal 𝑡 with respect to 𝑚 and 𝑙.
CORRESPONDING ANGLES: Let line 𝑡 be a transversal to lines 𝑙 and 𝑚. If ∠𝑥 and ∠𝑦 are alternate interior angles, and ∠𝑦 and ∠𝑧 are vertical angles, then ∠𝑥 and ∠𝑧 are corresponding angles.
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