Download Blame it All on Euclid

Blame it All on Euclid
Euclid was a Greek mathematician from the city of Alexandria, Egypt who first formalized geometry
in his work (13 books) entitled Elements. In it he begins with axioms (postulates). These are common
notions; basically things that are self-evident that everybody agrees are true. You might just call it
simple horse-sense that nobody in his right mind would argue about. For instance one of the first
postulates states that all right angles are equal to one another; pretty self-evident. Using these simple
postulates he goes about proving theorems (propositions or true statements). Once a theorem is proved
then it can be used to prove further theorems. These axioms and theorems then become the basis of
Euclidian Geometry, and thus Geometry students everywhere have Euclid to blame for “proofs”.
Some things are very easy to prove because they require only one theorem to prove them. For an
example, look below at the two lines cut by a transversal.
Congruent angles
Transversal line
Notice the two congruent angles are on the interior of the two lines, and in alternate positions with
respect to the transversal. These are called alternate interior angles. If one wanted to prove that the
two lines cut by the transversal were parallel, he or she would only need one existing theorem; “If two
lines are cut by a transversal so that the alternate interior angles are congruent, then those lines
are parallel.” That’s a pretty simple one. Most proofs require several steps and many theorems or
postulates to support them. For example, what if one were asked to prove that the diagonal of a
parallelogram divides that parallelogram into two congruent triangles (And he or she didn’t know of an
existing theorem to prove this? Below is a logical method, called the two-column method, that is
commonly used to perform the proof.
First begin by marking the given figure with everything that is known already. By definition the
opposite sides of a parallelogram are congruent, so mark them with tick marks. It is also a known
postulate that a segment is congruent to itself, so mark the diagonal with tick marks. This will make
writing the proof easier. Another postulate states that if two triangles each have the same measure of
sides then the two triangles are congruent (SSS). So if it can be first proved that the corresponding
sides of the parallelogram are congruent, then the SSS postulate can be used to prove the triangles are
congruent.
B
A
In the two-column method, list statements
on the left, with the proof for each
statement on the right. This is called the
deductive method, as you are coming to a
specific conclusion using a list of general
C
D
rules or concepts.
Reason (Proof)
Statement
segment AB  segment DC
segment AC  segment DB
Opposite sides of a parallelogram are congruent. (Side)
Opposite sides of a parallelogram are congruent. (Side)
segment BC  segment BC
Segment congruent to self. (Reflexive Property) (Side)
ΔABC  ΔDCB
By Side-Side-Side postulate (SSS)
Blame it All on Euclid
Notice to the right of the proof we included the word (Side) after each reason. This doesn’t have to
be included in a formal proof, but it is a helpful reminder to oneself as to the ultimate purpose of the
statement and reason. In this case since the SSS postulate was ultimately going to be used to prove the
congruence of the two triangles, it was necessary to first prove that the corresponding sides were
congruent. Using this method will help you to include only statements and reasons that are necessary
to make the proof.
While the deductive method begins with generally accepted statements (axioms) to prove specific
statements (theorems), the inductive method on the other hand goes from the specific to the general. It
is used in the scientific method where many observations are made of specific situations and then
general rules are made. It looks for patterns that can be explained by a general rule. One instance of
this in geometry is the triangle sum theorem. Although this theorem can be proved using the deductive
method, one could also go about carefully measuring the angles in hundreds of triangles and finding
the sum of the measures for each triangle. After coming up with the same answer (1800) for each
triangle one could reasonably conclude that all triangles have this same property. Of course if one
could find one triangle in which this wasn’t true (this is called a counterexample) then this conjecture
or hypothesis (a guess based upon some evidence) would be proved false. So for math, using induction
is not the strongest way to make a proof as one would need to prove every possible situation in order to
make the proof. For example, one would need to construct an infinite number of triangles with varying
measures of angles, measure each angle, and find the sum for the triangle to be absolutely sure that the
sum of the angles in any triangle is 1800. No one wants to waste his or her life doing that so it would be
far wiser to be like Euclid. He used the deductive method for his proofs, and so will we.
Blame it All on Euclid
For each of the following cases, decide whether deductive, or inductive reasoning was used.
1) Using theorems and postulates you prove that two triangles are congruent.
2) You make observations that after the leaves fall off the trees, it snows within a month. Therefore
you conclude that snow is caused by leaves falling off of trees.
3) You gather evidence from a crime scene and come to a logical conclusion that the butler did it.
4) At a school all the students were girls. All students were under the age of 18. Nancy was a student at
the school who was 16 years old. Therefore Nancy is a girl.
5) Josh noticed that whenever he had bananas on the counter for over a week that little fruit flies
appeared flying around the old bananas. Therefore he concluded that bananas were really fruit fly eggs.
6) A mathematician measured the diameters and the circumferences of 100 circles of different sizes.
He found that for each circle the ratio of the circumference to the diameter was 3.14159.
7) A mathematician knows that by definition all triangles have 3 sides. He also knows that by a
theorem that all triangles have angle sum measures of 1800. When given a polygon that fits this
criteria, he concludes it is a triangle.
8) Alex notices the answer of 2 x 2 is 4. He notices that 22 = 4. Therefore he concludes that 32 = 6.
9) Jill knows that equilateral triangles have angles of the same measure. She also knows that if the
measures of the corresponding angles in two triangles are all the same that the two triangles are at least
similar. She concludes then that equilateral triangles must be at least similar.
10) Brent notices that any time he measures the angles in a quadrilateral that the sum of the angles is
3600. Therefore he concludes that the sum of the angles in a quadrilateral is always 3600.