point that has integer coordinates, just to keep the problem simple and accurate. They will get the same distance no
matter where they measure though. An alternate definition of parallel lines is two lines that are a constant distance
apart.
Addition Exercises:
- Prove the Converse of the Perpendicular Transversal Theorem.
Answer: Refer to the figure at the top of page 178, at the beginning of this lesson.
TABLE2.6:
Statement Reason
←→
KNis perpendicular to←→
QT Given
←→
ORis perpendicular to←→
QT Given(^6) QPOis right Definition of Perpendicular Lines
(^6) PSTis right Definition of Perpendicular Lines
(^6) QPO∼= (^6) PST Right Angle Theorem
←→
ORis parallel to
←→
KN Converse of Corresponding Angle PostulateNon-Euclidean Geometry
Separate Worlds –The geometry presented in this section is completely separate from the geometry in the rest of the
text. The study of non-Euclidean geometry is excellent for developing critical thinking skills. It also demonstrates
to the students what an influential role postulates play and how important it is to carefully evaluate them before
accepting them as true. This section is best used for enrichment and should be treated differently from the other
sections. If the students attempt to memorize the postulates in this section it may compromise their ability to recall
analogous postulates of Euclidean Geometry. Exploring taxicab geometry is a wonderful way to spend a day in
class, but it is not something that has to be included on tests. This is a decision that the instruction can make based
on the ability of the students in a particular class.
Projects –This section opens the door to many possible projects that students can complete as part of the class
or for extra credit. More advanced students in particular will have the ability and interest to explore the topic of
Non-Euclidean geometry independently. Topics can include further exploration of taxicab geometry, other types of
Non-Euclidean geometry, like spherical geometry, or research into the mathematician who developed these fields.
This may make a good group project, where each group presents its findings to the class.
Encourage Creativity -Have students write their own problems involving taxicab geometry. This type of geometry
lends itself to application and story problems. Students can be creative and funny. They will enjoy sharing problems
with their classmates and solving each other’s challenges. Writing word application helps students solve similar
exercises. When formulating their question and deciding what information to give and how to give it, they become
more aware of the structure of a word problem. If the students are enjoying this line of study and there is time, they
may create their own type of geometry by setting up a system of postulates.
Abstraction and Modeling–This section briefly addresses the fact that mathematics is an abstraction and that it
usually needs to be modified before it can be helpful in applications to the world in which we live. This is an
important concept applicable to all areas of mathematics that is easily seen while studying geometry. This knowledge
will help students understand why math is useful and how they will benefit from what they are learning in this class.
2.3. Parallel and Perpendicular Lines