§7.1 Quadrilaterals
The student will learn:
the definition of various quadrilaterals
and theorems associated
with quadrilaterals.
1
Quadrilateral
Definition
Let A, B, C, and D be four coplanar points. If no
three of these points are collinear, and the
segments AB, BC, CD. And DA intersect only at
their end points, then the union of the four
segments is called a quadrilateral. The segments
are the sides and the points are the vertices.
Segments AC and BD are the diagonals.
2
Rectangle & Square
Definitions
If all four angles of a quadrilateral are right
angles, then the quadrilateral is a rectangle.
If all four angles of a quadrilateral are right
angles and all sides are congruent, then the
quadrilateral is a square.
3
Parallelogram & Trapezoid
Definitions
A parallelogram is a quadrilateral in which both
pairs of opposite sides are parallel.
A trapezoid is a quadrilateral in which exactly
one pair of opposite sides are parallel. The
parallel sides are called the bases.
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Note
Some of the previous definitions have classified
quadrilaterals by sides and/or angles and/or
diagonals. You will be asked in homework to
organize quadrilaterals using those
specifications.
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Before we continue our study of
quadrilaterals we will need the
following information on parallelism.
Historical Background
Euclid’s Fifth. If a straight line falling on two
straight lines makes the interior angles on the
same side less than two right angles, the two
straight lines, if produced indefinitely, meet on
that side which are the angles less than the two
right angles.
n
l
A
m
 1 +  2 < 180 then
lines l and m meet on
the A side of the
transversal n.
7
Playfair’s Postulate
Given a line l and a point P not on l, there exist
one and only one line m through P parallel to l.
8
Equivalent Forms of the Fifth
 Area of a right triangle can be infinitely large.
 Angle sum of a triangle is 180.
 Rectangles exist.
 A circle can pass through three points.
 Parallel lines are equidistant.
 Given an interior point of an angle, a line can be
drawn through the point intersecting both sides of
the angle.
9
Euclidean Parallelism
Definition. Two distinct lines l and m are said
to be parallel, l || m, iff they lie in the same
plane and do not meet.
10
Theorem 1: Parallelism in
Absolute Geometry
If two lines in the same plane are cut by a
transversal so that a pair of alternate interior
angles are congruent, the lines are parallel.
Notice that this is a theorem and not an axiom
or postulate.
11
Parallelism in Absolute Geometry
Given: l, m and transversal t. and
1≅ 2
Proof by contradiction.
(1) l not parallel to m, meet at R.
Prove: l ‫ ׀׀‬m
(2)  1 is exterior angle
Assumption
Def
(3) m  1 > m  2
Exterior angle inequality
(4) → ←
Given  1 ≅  2
l
t
A
1
m
2
B
R
12
Theorem
If parallel lines are cut by a transversal then the
following relationships are true. (Without proof.)
•Alternate interior angles are equal (3 & 6, 4 & 5.)
•Alternate exterior angles are equal (1 & 8, 2 & 7.)
•Corresponding angles are equal. (1 & 5, 2 & 6, 3 & 7,
4 & 8.)
1
3
5
7
2
4
6
8
13
Theorem
Each diagonal separates a parallelogram into
two congruent triangles. Do you see the proof.
14
Theorem
In a parallelogram, any two opposite sides are
congruent.
In a parallelogram, any two opposite angles are
congruent.
15
Theorem
In a parallelogram, the diagonals bisect each
other.
In a parallelogram, any two consecutive angles
are supplementary.
Prove!
16
Theorem
A quadrilateral in which both pairs of opposite
sides are congruent is a parallelogram.
If two sides of a quadrilateral are parallel and
congruent, then it is a parallelogram.
Prove!
If the diagonals of a quadrilateral bisect each
other, then it is a parallelogram.
17
Theorem
If a parallelogram has one right angle, then it
has four right angles, and is a rectangle.
In a rhombus the diagonals are perpendicular to
one another.
Prove!
If the diagonals of a quadrilateral bisect each
other and are perpendicular, then it is a
rhombus.
18
Theorem
Quadrilaterals may be proven congruent by one
of the following means:
SASAS
ASASA
SASAA
SASSS
The proof is left to the student.
19
Theorem – The Midline Theorem
The segment between the midpoints of two sides
of a triangle is parallel to the third side and half
as long.
Hint:
B
w
D
y
E
F
x
v
A
C
20
Cyclic Quadrilaterals
Definitions
A quadrilateral is cyclic if its four vertices lie on
a circle.
21
Theorems
The perpendicular bisectors of the four sides of
a cyclic quadrilateral are concurrent.
The perpendicular bisectors of the two
diagonals of a cyclic quadrilateral are
concurrent at the center of the circle.
The opposite angles of a cyclic quadrilateral are
supplementary.
Sketchpad on Campus
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Assignment: §7.1
Add HW on parallelism