Download Counter Examples To show that a statement is false it is enough to

Counter Examples
To show that a statement is false it is enough to show
one example that satisfies the conditions of the statement, but for which the conclusion of the statement
does not hold.
For example, to show that the statement “All four sided
shapes in the plane are rectangles.” is false it is enough
to show one non-right angled parallelogram.
Such an example is called a counter example to the
claim or statement.
Example. Use a counter example to show that the
statement “Real numbers greater than 10 are all divisible by 5.” is not in general true.
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Contradiction
Sometimes it is easiest to prove something by first assuming that it is not true, then by using that assumption
arriving to some statement that we already know to be
false.
To effectively use a proof by contradiction we need to
be able to clearly determine the negation of the original
statement.
The negation on an “If P then Q.” statement is “P and
not Q.”.
Example. Write a negation of the statement: If the
shape has three sides, then it is a triangle.
Example. Use a proof by contradiction to show that
if m is an even integer then m − 5 must be odd.
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1.4 More about Congruency
Theorem 1.4.1, SAA congruency: Two triangles are
congruent if two angles and a side of one are congruent
to two angles and a side of the other.
To show that there is no general SSA congruency rule
for triangles we provide a counter example.
If we add an extra condition we can make an SSA rule.
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Theorem 1.4.2, SSA∗ : 4ABC ≡ 4DEF if AB ≡ DE,
AC ≡ DF , ∠B ≡ ∠E and AC ≥ AB.
Proof:
Corollary 1.4.3, HSR Congruency: If the hypotenuse
and one side of a right triangle is congruent to the hypotenuse and one side of another triangle, then the two
triangles are congruent.
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Perpendiculars & Angle Bisectors
Definition. Two lines that intersect each other at right
angles are perpendicular to each other. The right bisector or perpendicular bisector of a line segment
AB is a line that is perpendicular to AB, intersecting
the midpoint M of AB.
Theorem 1.4.5, Characterization of the Perpendicular Bisector: Given different points A and B, the perpendicular bisector of AB consists of all points P that
are equidistant to A and B.
Proof:
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−−→
Definition. Given a non-reflex angle ∠ABC, a ray BD
such that ∠ABD = ∠CBD is called a angle bisector of
∠ABC.
Definition. Given a line l and a point P not on l, the
distance from P to l, denoted d(P, l), is the length of
the segment P Q where Q is the point where the line
through P and perpendicular to l meets l.
Theorem 1.4.10, Characterization of the Angle Bisector: The angle bisector of a non-reflex angle θ consists of all points interior to theta that are equidistant
from the arms of the angle.
Proof:
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