formulaic. Students who are struggling with proofs can get some practice with this style of writing while already
knowing the structure of the proof.
1
stStatetheıi fsideincongruence f orm.
2
ndChangethecongruenceo f segmentsintoequalityo f numbers.
3
rdA p plytheanalogous pro pertyo f equality.
4
thChangetheequalityo f numbersbacktocongruenceo f segments.
Theorems –The concept of a theorem and how it differs from a postulate has been briefly addressed several times
in the course, but this is the first time theorems have been the focus of the section. Now would be a good time for
students to start a theorem section in their notebook. As they prove, or read a proof of each theorem it can be added
to the notebook to be used in other proofs.
Additional Exercises:
- Prove the following statement.
IfAB=AC, triangleABCis isosceles.
Answer:
TABLE2.1:
Statement Reason
AB=AC Given
AB∼=AC Definition of congruent segments.
TriangleABCis isosceles. Definition of isosceles triangle.Proofs About Angle Pairs
Mark-Up That Picture –Angles are sometimes hard to see in a complex picture because they are not really written
on the page; they are the amount of rotation between two rays that are directly written on the page. It is helpful
for students to copy diagram onto their papers and mark all the angles of interest. They can use highlighters and
different colored pens and pencils. Each pair of vertical angles or linear pairs can be marked in a different color.
Using colors is fun, and gives the students the opportunity to really analyze the angle relationships.
Add New Information to the Diagram –It is common in geometry to have multiple questions about the same
diagram. The questions build on each other leading the student though a difficult exercise. As new information is
found it should be added to the diagram so that it is readily available to use in answering the next question.
Try a Numerical Example –Sometimes students have trouble understanding a theorem because they get lost in all
the symbols and abstraction. When this happens, advise the students to assign a plausible number to the measures of
the angles in question and work form there to understand the relationships. Make sure the student understands that
this does not prove anything. When numbers are assigned, they are looking at an example, using inductive reasoning
to get a better understanding of the situation. The abstract reasoning of deductive reasoning must be used to write a
proof.
Inductive vs. Deductive Again –The last six sections have given the students a good amount of practice drawing
Chapter 2. Geometry TE - Common Errors