Download Feb 17 Notes: Definition: If A, B, C and D are any four points lying in

Feb 17 Notes:
Definition: If A, B, C and D are any four points lying in a plane
such that no three of them are collinear, and if the points are so
situated that no pair of the segments
and
intersect at an interior point, then the set ABCD =
is a quadrilateral. The vertices are
the points A, B, C, and D. The sides are
, and
. The diagonals are
and
pDAB, pABC, pBCD, and pCDA.
. The angles are
Sides with a common endpoint are called adjacent or
consecutive. Angles containing a common side are also called
adjacent or consecutive. Two sides or angles that are not
adjacent are called opposite.
Examples:
‘ABCD, ‘EFGH, and ‘IJLK are quadrilaterals; but the figure
formed by points M, N, O, and P is not.
are diagonals.
and
are adjacent sides, while
opposite sides.
pJIL and pILK are adjacent angles
pEFG and pFGH are adjacent angles
pFEH and pFGH are opposite angles
pABC and pADC are opposite angles
and
are
Notice that in ‘ABCD:
1.
2.
3.
4.
The diagonals intersect.
Vertices are interior to their opposite angles.
Sides lie entirely in one half-plane of their opposite sides.
The diagonals lie between opposite vertices.
Notice that in ‘EFGH, each of these is violated. We will call
quadrilaterals like ‘ABCD convex. But first, we need a
theorem.
Theorem: For a given quadrilateral, the following are
equivalent:
1. Each side of the quadrilateral is on a halfplane of the
opposite side of the quadrilateral.
2. The vertex of each angle of the quadrilateral is in the interior
of its opposite angle.
3. The diagonals intersect each other (at interior points).
4. The diagonals lie between opposite vertices (i.e. opposite
vertices are on opposite sides of the diagonal).
Proof: By the excited and clever students of Math 362.
We take any of the equivalent statements in the previous
theorem as the definition of a convex quadrilateral.
Definition: Two quadrilaterals ‘ABCD and ‘XYZW are
congruent under the correspondence ABCD : XYZW iff all
pairs of corresponding sides and angles under the
correspondence are congruent. In other words, this is another
application of the basic definition abbreviated by CPCF.
To no one’s surprise, we abbreviate this by
‘ABCD – ‘XYZW.
Theorem: SASAS Congruence. Suppose that two convex
quadrilaterals ‘ABCD and ‘XYZW are such that, under the
correspondence ABCD : XYZW, three consecutive sides and
the two angles included by those sides of ‘ABCD are
congruent, respectively, to the corresponding three consecutive
sides and two included angles of ‘XYZW. Then ‘ABCD –
‘XYZW.
Proof: Draw a pair of corresponding diagonals in the two
quadrilaterals and use congruence criteria for triangles.
Convexity allows us to add the angles when we need to, and the
problem reduces to congruence for triangles.
Other Similarly Proved theorems:
ASASA Theorem
SASAA Theorem
SASSS Theorem
Question: What about ASAA? SSSS?
Definintion: A rectangle is a convex quadrilateral having four
right angles.
We would like rectangles to exist. In fact, we cannot prove that
they do exist yet. The nearest we can come is a somewhat
strange creature called a Saccheri Quadrilateral.
Definition: Let
be any line segment, and erect two
perpendiculars at the endpoints A and B. Mark off points C and
D on these perpendiculars so that C and D lie on the same side
of the line
, and BC = AD. Join C and D. The resulting
quadrilateral is a Saccheri Quadrilateral. Side
is called
the base,
and
the legs, and side
the summit. The
angles at C and D are called the summit angles.
Lemma: A Saccheri Quadrilateral is convex.
Proof: By construction, D and C are on the same side of the line
, and by PSP,
intersected
will be as well. But then if
at a point F, one of the triangles ªADF or
ªBFC would have two angles of at least measure 90, a
contradiction (one of the linear pair of angles at F must be
obtuse or right). Since perpendiculars to a line are unique,
and
cannot meet at all, so
and
must lie entirely
on one side of each other. Thus ‘ABCD is convex.
Theorem: The summit angles of a Saccheri Quadrilateral are
congruent.
Proof: The SASAS version of using SAS to prove the base
angles of an isosceles triangle are congruent. ‘DABC –
‘CBAD, by SASAS, so pD – pC.
Corollaries:
•
The diagonals of a Saccheri Quadrilateral are congruent.
(Proof:
ªABC – ªBAD by SAS; CPCF gives AC = BD.)
•
The line joining the midpoints of the base and summit of a
quadrilateral is the perpendicular bisector of both the base
and summit. (Proof: Let N and M be the midpoints of
summit and base, respectively. Use SASAS on ‘NDAM
and ‘NCBM. The angles at M and N are congruent by
CPCF, and form a linear pair, so must have measure 90.)
•
If each of the summit angles of a Saccheri Quadrilateral is
a right angle, the quadrilateral is a rectangle, and the
summit is congruent to the base. (Proof: Consider diagonal
. HL gives ªADC – ªCBA so DC = AB.)