Download Alternate Interior Angles Terminology: When one line t intersects

Alternate Interior Angles
Terminology: When one line t intersects each of two given lines l and
m at distinct points, we call that line a transversal. We often say that
lines l and m are cut by the transveral t.
We define alternate interior angles, corresponding angles, alternate
exterior angles, and interior angles on the same side of the transversal
using various betweeness and half-plane notions.
Definitions: Suppose line l intersects lines m and n at points B and E,
respectively, with points A and C on line m and points D and F on line
n such that A*B*C and D*E*F, with A and D on the same side of l.
Suppose also that G and H are points such that H*E*B and E*B*G,
Then:
•
pABE and pBEF are alternate interior angles, as are pCBE
and pDEB.
•
pABG and pFEH are alternate exterior angles, as are pCBG
and pDEH.
•
pGBC and pBEF are a pair of corresponding angles, as are
pGBA & pBED, pCBE & pFEH, and pABE & pDEH.
•
pCBE and pFEB are interior angles on the same side of the
transversal, as are pABE and pDEB.
Alternate Interior Angles Theorem: If two lines n and m are cut by a
transversal l so that a pair of alternate interior angles are congruent, the
lines are parallel.
~ Let l intersect lines m and n at points A and B respectively. Let
p1– p2. Suppose for contradiction that m and n meet at point C.
Then either p1 is exterior to ªABC, or p2 is exterior to ªABC. In the
first case, the exterior angle inequality gives µ(p1) > µ(p2); in the
second, it gives µ(p2) > µ(p1). In either case, we have a contradiction
to p1– p2. €
Note: It is important to distinguish between the Alternate Interior
Angles Theorem and its converse. The Alternate Interior Angles
Theorem can be proved in neutral geometry. Its converse is actually
equivalent to Euclid’s parallel postulate and so cannot be proved in
neutral geometry, as we will discuss in detail later.
Corollary (Corresponding Angles Theorem): If two lines n and m
are cut by a transversal l so that a pair of corresponding angles are
congruent, the lines are parallel.
~ Follows immediately by using vertical angles. €
Corollary: If two lines n and m are cut by a transversal l so that a pair
of interior angles on the same side of the transversal are
supplementary, the lines are parallel.
~ Follows immediately by using the Linear Pair Theorem. €
Corollary (Existence of Parallels): Given a line l and an external
point P, there is a line m such that P lies on m and m is parallel to l.
~ Drop a perpendicular from P to l; call the foot of that perpendicular
Q. Let
. Now, through point P use the protractor postulate to
construct a line perpendicular to t. Now, viewing t as a transversal for
l and m, all the alternate interior angles are right angles, and l and m
must be parallel. ~
Note that this construction produces a parallel line and a transversal
that is perpendicular to both lines. We will use this construction
frequently to obtain parallel lines through an external point.
Recall the three parallel postulates:
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.
Elliptic Parallel Postulate: For every line l and for every point P that
does not lie on l, there is no line m such that P lies on m and m is
parallel to l.
Hyperbolic Parallel Postulate: For every line l and for every point P
that does not lie on l, there at least two lines m and n such that P lies
on both m and n and both m and n are parallel to l.
We can now eliminate one of these three parallel postulates, since we
are able to construct for any line l and any exterior point P at least one
parallel line through P. We state this and another useful fact in two
easy corollaries.
Corollary: The Elliptic Parallel Postulate is false in any model of
neutral geometry.
Corollary: If l, m, and n are any three lines such that
, then either
.