Euclid's Fifth Postulate
Definition (Parallel Lines) Two distinct lines
and
are said to be parallel when they lie in the same plane and they do not meet.
Definition (Transversal) If lines
and
all lie in the same plane and
intersects
and
then
is called a transversal of lines
and
Theorem (Parallelism) If two lines in the same plane are cut by a transversal so that a pair of alternating interior angles are congruent, then the lines are parallel.
Proof. We are given two lines
and
meeting line
at points
and
respectively.
We also are given a pair of alternating interior angles, say
and
with
Assume for a contradiction, that
is not parallel to
By definition of parallel lines,
and
meet at some point
By definition of exterior angles,
is an exterior angle of
By the Exterior Angle Inequality Theorem,
since
We have reached the contradiction:,
and
because of the trichotomy property of the real numbers.
Therefore, in fact
and
must be parallel.
Axiom (Euclid's Fifth Postulate) If
is any line and
is any point not on
there exists in the plane of
and
one and only one line
that passes through
and is parallel to
Theorem (Transversal Properties) Here are some properties of transversals.
(i) If two parallel lines are cut by a transversal, then each pair of alternate interior angles are congruent.
(ii) If two lines in the same plane are cut by a transversal, then the two lines are parallel if and only if a pair of interior angles on the same side of the transversal are supplementary. (iii) If two lines in the same plane are cut by a transversal, then the two lines are parallel if and only if a pair of corresponding angles are congruent. (iv) If two lines in the same plane are cut by a transversal, then the lines are parallel if and only if a pair of alternating interior angles are congruent. (v) If a line is perpendicular to one of two parallel lines, it is perpendicular to the other also.
Theorem (Euclidean Exterior Angle)
(i) The measure of an exterior angle of any triangle equals the sum of the measure of the two opposite interior angles.
(ii) The sum of the measures of the angles of any triangle is 180.
Proof.
Say we are given triangle
with exterior angle
we can construct segment
so that the midpoint
of
is also the midpoint of
.
Since
and
we know that
by
Therefore,
Then using the above properties,
![euclids fifth postulate _gr_53.gif]](pages/euclids-fifth-postulate/Images/euclids-fifth-postulate_gr_53.gif)
Theorem (Midpoint Connector)
(i) The segment joining the midpoints of two sides of a triangle is parallel to the third side and has length one-half that of the third side.
(ii) If a line bisects one side of a triangle and is parallel to the second, it also bisects the third side.
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Cite this as: Euclids Fifth Postulate Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/euclids-fifth-postulate.html
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