Set up a proof for the following information then complete the proof. Given: Two isosceles triangles have the same base. Prove: The line...

We are given
two isosceles triangles with the same base. We are asked to prove that the line joining the
vertex angles is the bisector of the base.

When asked to do a proof, there
are a few general steps that you can follow.

(1) If you
are not provided with a diagram you should draw one. Note that you do not want your diagram to
be "special" (e.g., do not draw right or equilateral triangles unless that is
given).

(2) List the given information and mark your diagram
accordingly.

(3) Write down what you are to prove. Make sure you understand
the problem, and you might try to see if you believe the conclusion is true.

(4) Write down any inferences you can get from the diagram or the givens. (I try to use
a different color for information that is given and facts that I have deduced.)

(5) Determine the conditions under which the conclusion is true and ascertain if you
have enough information.

(1) Draw the diagram. Label one
of the isosceles triangles ABC (with A the vertex angle.) Then using BC as the base, draw
another isosceles triangle with D as the vertex. (Note that D could be on either side of BC. If
D is on the same side of BC as A you will have a triangle inside of a triangle. If you place D
on the opposite side, you will have a kite. The proof is the same for both cases so choose one
and note that there is no loss of generality.)

(2) We have AB = AC and DB =
DC, so we mark the diagram accordingly. Since we are asked to prove something about the line
joining the vertices, draw that segment also. (Through two points there is exactly one
line.)

(3) / (4) Label the point of intersection of the line through AD and
BC as E. Assuming we know the isosceles triangle theorem and its converse, we see that `/_ ABC
cong /_ACB, /_DBC cong /_DCB`, and we mark those accordingly.

(5) For AD to
bisect BC we must have BE = EC. This can be shown as these are corresponding segments of
congruent triangles.

One idea for the proof:

a. List the givens.
b. Note that AB = AC, DB = DC by the definition of
isosceles triangles.
c. Note that the base angles of triangles ABC and DBC are
congruent by the isosceles triangle theorem.
d. Since AD = AD (reflexive property), we
have triangle ADB congruent to triangle ADC by SSS.
e. Thus angle BAD is congruent to
angle CAD by CPCTC (corresponding parts of congruent triangles are congruent.)
f. Now
since AE = AE, we also have triangle ABE congruent to triangle ACE by SAS
g. Finally,
BE is congruent to EC by CPCTC.
h. By definition, AD is the bisector of BC.

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