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Common Core For Geometry

Worksheets for Geometry, Module 1, Lesson 18

Student Outcomes

- Students learn to construct a line parallel to a given line through a point not on that line using a rotation by 180Λ. They learn how to prove the alternate interior angles theorem using the parallel postulate and the construction.

**Looking More Carefully at Parallel Lines**

Classwork

**Opening Exercise**

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?

**Discussion**

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.

**Example 1**

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:

- Suppose a transversal intersects a pair of lines. If two alternate interior angles are equal in measure, then the pair of lines are parallel.
- Suppose a transversal intersects a pair of lines. If the lines are parallel, then the pair of alternate interior angles are equal in measure. The second part turns out to be an equivalent form of the parallel postulate. To build up to the theorem, first we need to do a construction

**Example 2**

Given a line π and a point π not on the line, follow the steps below to rotate π by 180Β° to a line πβ²
that passes through π:

a. Label any point π΄ on π.

b. Find the midpoint of segment π΄π using a ruler. (Measure the length of segment π΄π, and locate the point that
is distance
π΄π/2
from π΄ between π΄ and π.) Label the midpoint πΆ.

c. Perform a 180Β° rotation around center πΆ. To quickly find the image of π under this rotation by hand:

i. Pick another point π΅ on π.

ii. Draw πΆπ΅.

iii. Draw circle: center πΆ, radius πΆπ΅.

iv. Label the other point where the circle intersects πΆπ΅ by π.

v. Draw ππ.

d. Label the image of the rotation by 180Β° of π by πβ² = π
_{πΆ,180}(π).

How does your construction relate to the geometry assumption stated above to rotations? Complete the statement
below to clarify your observations:

π
_{πΆ,180} is a 180Β° _____ around πΆ. Rotations preserve ____ ; therefore π
_{πΆ,180}, maps the line π to the
line ____ . What is R_{πΆ,180}(π΄)?

**Example 3**

The lines π and πβ² in the construction certainly look parallel, but we do not have to rely on looks.

Claim: In the construction, π is parallel to πβ².

**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 π. Since π is on πβ², it must be the image of some point π on π under the π
_{πΆ,180}
rotation, (i.e., π
_{πΆ,180}(π) = π). Since π
_{πΆ,180} is a 180Β° rotation, π and π must be the endpoints of a diameter of a circle
that has center πΆ. In particular, ππ must contain πΆ. Since π is a point on π, and π is a different point on π (it was the
intersection of both lines), we have that π = ππ because there is only one line through two points. But ππ also contains
πΆ, which means that π contains πΆ. However, πΆ was constructed so that it was not on π. This is absurd.

There are only two possibilities for any two distinct lines π and πβ² 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 π and πβ² were parallel to begin with.

**Example 4**

The construction and claim together implies the following theorem.

**THEOREM**: Given a line π and a point π not on the line, then there exists line πβ²
that contains π and is parallel to π.

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.

**Example 5**

We are now ready to prove the first part of the basic fact above. We have two lines, π and πβ², and all we know is that a transversal π΄π intersects π and πβ² such that a pair of alternate interior angles are equal in measure. (In the picture below, we are assuming πβ πππ΄ = πβ π΅π΄π.)

Let πΆ be the midpoint of π΄π. What happens if you rotate 180Β° around the center πΆ? Is there enough information to
show that π
_{πΆ,180}(π) = πβ²?

a. What is the image of the segment π΄π?

b. In particular, what is the image of the point π΄?

c. Why are the points π and π
_{πΆ,180}(π΅) on the same side of π΄π ?

d. What is the image of π
_{πΆ,180}(β π΅π΄π)? Is it β πππ΄? Explain why.

e. Why is π
_{πΆ,180}(π) = πβ²?

We have just proven that a rotation by 180Β° takes π to πβ². By the claim in Example 3, lines π and πβ² 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.

**Discussion**

In Example 5, suppose we had used a different rotation to construct a line parallel to π that contains π. Such constructions are certainly plentiful. For example, for every other point π· on π, we can find the midpoint of segment ππ· and use the construction in Example 2 to construct a different 180Β° rotation around a different center such that the image of the line π is a parallel line through the point π. Are any of these parallel lines through π different? In other words,

Can we draw a line other than the line πβ² through π that never meets π?

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 π in the plane not lying on a line β, every line in the plane that contains π intersects β except at most one lineβthe one we call parallel to β.

**Example 6**

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 π΄π intersects line π at π΄ and πβ² at π, pick and label another point π΅ on π, and choose a point π on πβ² on the opposite side of π΄π as π΅. The picture might look like the figure below:

Let πΆ be the midpoint of π΄π, and apply a rotation by 180Β° around the center πΆ. As in previous discussions, the image of
π is the line π
_{πΆ,180}(π), which is parallel to π and contains point π. Since π
β²
and π
_{πΆ,180}(π) are both parallel to π and contain
π, by the parallel postulate, they must be the same line: π
πΆ,180(π) = π
β²
. In particular, π
_{πΆ,180}(β π΅π΄π) = β πππ΄. Since
rotations preserve angle measures, πβ π΅π΄π = πβ πππ΄, which was what we needed to show.

**Discussion**

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.

**Example 7**

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 π1
, π2
, and π3
in the plane have the property that π1 β₯ π2 and π2 β₯ π3
, then π1 β₯ π3
. (In
proofs, this can be written as, βIf two lines are parallel to the same line, then they are parallel to each other.β)

**Relevant Vocabulary**

**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 π and π in a plane, a third line π‘ is a transversal if it intersects π at a single point and
intersects π at a single but different point.
The definition of transversal rules out the possibility that any two of the lines π, π, and π‘ are the same line.

**ALTERNATE INTERIOR ANGLES**: Let line π‘ be a transversal to lines π and π such that π‘ intersects π at point π and intersects π
at point π. Let π
be a point on π and π be a point on π such that the points π
and π lie in opposite half planes of π‘. Then
the β π
ππ and the β πππ are called alternate interior angles of the transversal π‘ with respect to π and π.

**CORRESPONDING ANGLES**: Let line π‘ be a transversal to lines π and π. If β π₯ and β π¦ are alternate interior angles, and β π¦
and β π§ are vertical angles, then β π₯ and β π§ are corresponding angles.

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