Download Basic Theorems of Euclidean Geometry We finally adopt the

Basic Theorems of Euclidean Geometry
We finally adopt the Euclidean Parallel Postulate and explore
geometry under this assumption. The result is Euclidean Geometry
and its major results are similarity, the Pythagorean Theorem, and
trigonometry, although there is of course much more.
Axiom (The Euclidean Parallel Postulate): For every line l and for
every point P that does not lie on l, there is exactly one line m such that
P lies on m and m is parallel to l.
We now have as theorems all those statements that we proved
equivalent to EPP earlier, including:
1.
If two parallel lines are cut by a transversal, alternate interior
angles are congruent.
2.
(Euclid’s Postulate V) If two lines in the same plane are cut by
a transversal so that the sum of the measures of a pair of
interior angles on the same side of the transversal is less than
180, then the lines will meet on that side of the transversal.
3.
(Proclus’s Axiom) A third line intersecting one of two parallel
lines intersects the other.
4.
A line perpendicular to one of two parallel lines is
perpendicular to the other.
5.
The perpendicular bisectors of the sides of a triangle are
concurrent.
6.
There exists a circle passing through any three noncollinear
points.
7.
(Angle Sum Postulate) The sum of measures of the angles of a
triangle is 180.
8.
There exists one triangle such that the sum of measures of the
angles is 180.
9.
There exists a rectangle.
10.
There exist two lines l and m such that l is equidistant from m.
11.
There exists a pair of similar, noncongruent triangles.
12.
There exists an acute angle such that every line intersecting and
perpendicular to one ray of the angle intersects the other ray.
13.
(Wallis’ Postulate) If ªABC is a triangle and
is a
segment, then there exists a point F such that ªABC is similar
to ªDEF.
14.
(Transitivity of Parallelism) If line l is parallel to line m and
line m is parallel to line n, then either n = l or l is parallel to n.
There are also a flood of straightforward corollaries.
Theorem: If two lines in the same plane are cut by a transversal, then
the lines are parallel iff
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alternate interior angles are congruent
alternate exterior angles are congruent
corresponding angles are supplementary
interior angles on the same side of the transversal are
supplementary
exterior angles on the same side of the transversal are
supplementary
Note: Now we begin to strengthen some of the theorems of absolute
geometry, beginning with the Exterior Angle Inequality.
Euclidean Exterior Angle Theorem: In any triangle, the measure of
an exterior angle is the sum of the measures of the two remote interior
angles.
~ We use the same construction as for the proofs of the exterior angle
theorem and Saccheri - Legendre Theorem in absolute geometry.
Begin with ªABC and point D with B*C*D. Find the midpoint M of
and find point E with B-M-E and MB = ME. Connect E and C.
Now by construction and SAS, ªBAM is congruent to ªECM, and it is
easy to show that E is interior to pACD. Moreover, since pBAC
–pECA, with
as the transversal we have congruent alternate
interior angles and thus
. This makes pABC and pECD
corresponding angles (with
congruent. Thus
as the transversal), and hence
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Definitions:
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A convex quadrilateral is a parallelogram if the opposite sides
are parallel.
A rhombus is a parallelogram having two adjacent sides
congruent.
A square is a rhombus having two adjacent sides
perpendicular.
A cascade of theorems, all of which rely on cutting things into
triangles and using congruence theorems and transversal theorems:
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A diagonal of a parallelogram divides it into congruent
triangles.
A convex quadrilateral is a parallelogram iff its opposite sides
are congruent.
Opposite angles of a parallelogram are congruent
Adjacent angles of a parallelogram are supplementary
A convex quadrilateral is a parallelogram iff its diagonals
bisect each other.
A parallelogram is a rhombus iff its diagonals are
perpendicular.
A parallelogram is a rectangle iff its diagonals are congruent.
A parallelogram is a square iff its diagonals are both
perpendicular and congruent.
Etc. These make good exercises. Try a few!!
Definition: A trapezoid is a convex quadrilateral with exactly one
pair of opposite sides parallel. The parallel sides are called bases and
the other two sides are called legs. The segment joining the midpoint
of the legs is called the median.
Some books use a definition of trapezoid that omits the word
“exactly.” This allows trapezoids to be parallelograms; that is, the two
legs could also be parallel. For us, the set of all trapezoids and the set
of all parallelograms are disjoint.
If a trapezoid is such that its two legs are congruent, it is called an
isosceles trapezoid.
Can you prove that the base angles of an isosceles trapezoid are
congruent?