If they are having trouble, a review of the Deductive Reasoning section in Chapter Two: Reasoning and Proof will
help. It could be assigned as reading the night before the current lesson will be done in class.
This is a good point to summarize what the students have learned in this chapter about congruent triangles and
demonstrate how it can be put to use. To understand this proof, students need to remember that the definition of
congruent triangles requires three pairs of congruent sides and three pairs of congruent angles, but realize that not all
six pieces of information need to be verified before it is certain that the triangles are congruent. There are shortcuts.
The proof of the Base Angle Theorem uses one of these shortcuts and jumps to congruence which implies that the
base angles, a pair of corresponding angles of congruent triangles, are congruent.
To a student new to geometry this argument is not as straightforward as it may seem to an instructor experienced in
mathematical proofs. Plan to take some time explaining this important proof.
A Proved Theorem Can Be Used –Now that the students have the proof of the Base Angle Theorem they can use
it as opportunities present themselves. They should be on the lookout for isosceles triangles in the proofs of other
theorems, in complex figures, and in all other situations. When they spot them, they need to immediately apply the
Base Angle Theorem and mark those base angles congruent. This is true for the converse as well. When they spot a
triangle with congruent angles, they should mark the appropriate sides congruent. Students sometimes do not realize
what a powerful tool this theorem is and that they will be using it extensively throughout this class, and in math
classes they will take in the future.
Additional Exercise:
- What are the measures of the angles of an equilateral triangle? What postulates or theorems did you use to obtain
your answer?
Answer: 60 degrees, Base Angle Theorem, Triangle Sum Theorem
Congruence Transformations
Reflection or Rotation –When looking a two triangle, where one is a transformation of the other, students some-
times have trouble distinguishing between a reflection and a rotation. This is particularly true when the triangles
are almost equilateral. When demonstrating these transformations, it is best to use an obviously scalene triangle.
Good use of labels is also helpful. The prime notation clearly indicates the new location of each vertex under the
transformation. A rotation preserves the order of the vertices, and a reflection reverses the order of the vertices. If
the students are unsure of what transformation has been applied to the figure, have them choose one vertex and then
move counterclockwise around the polygon listing off the vertices as they occur. If they start with the image of that
first vertex in the new figure, and again move counterclockwise, they will get the images of the vertices in the same
order for a rotation, and in reverse order for a reflection.
Don’t Just Memorize, Reason –This section contains many ordered pair rules for different transformation. Stu-
dents will try to memorize them without really thinking about them or looking for patterns. This is challenging, if
not impossible for most students. Have the students discuss similarities and differences between the rules. Ask them
if the rule surprises them, or seems logical. Why? If they really get stuck, they can do a test. Graph a scalene triangle
and apply different rules to it until the desired transformation occurs. Students will be motivated to use reason to
shorten the guess and check process. In geometry problems are written so that students will have to think about
them for awhile, and figure out an answer. Once students realize that they are not supposed to know the answer
immediately, they are much more willing to spend time thinking about an exercise.
Additional Exercise:
- Graph a scalene triangle in the first quadrant of the coordinate axis. Reflect the triangle over thex−axis. Take
this new triangle and reflect it over they−axis. What single transformation would take the first triangle to the final
triangle? How can this be predicted by the ordered pair rules?
Chapter 2. Geometry TE - Common Errors