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Quadrilateral Family
Topic Index | Geometry Index | Regents Exam Prep Center
Each member of the quadrilateral family will describe its specific properties.
*Quadrilateral
I have exactly four sides.
The sum of the interior
angles of all quadrilaterals
is 360º.
*Trapezoid
I have only one set of parallel
sides.
[The median of a trapezoid is
parallel to the bases and equal to
one-half the sum of the bases.]
A quadrilateral is any four sided
figure. Do not assume any additional
properties for a quadrilateral unless
you are given additional information.
A trapezoid has ONLY
ONE set of parallel
sides. When proving a
figure is a trapezoid, it
is necessary to prove
that two sides are
parallel and two sides
are not parallel.
*Isosceles
Trapezoid
I have:
- only one set of parallel sides
- base angles congruent
- legs congruent
- diagonals congruent
- opposite angles supplementary
*Parallelogram
Never assume that a
trapezoid is
isosceles unless you
are given (or can
prove) that
information.
Notice how the properties
of a parallelogram come
in sets of twos: two
properties about the sides;
two properties about the
angles; two properties
about the diagonals. Use
this fact to help you
remember the properties.
I have:
- 2 sets of parallel sides
- 2 sets of congruent sides
- opposite angles congruent
- consecutive angles
supplementary
- diagonals bisect each other
- diagonals form 2 congruent
triangles
*Rectangle
I have all of the properties of the
parallelogram PLUS
- 4 right angles
- diagonals congruent
If you know the
properties of a
parallelogram, you
only need to add 2
additional properties to
describe a rectangle.
*Rhombus
A rhombus is a slanted
square. It has all of the
properties of a parallelogram
plus three additional
properties.
I have all of the properties of the
parallelogram PLUS
- 4 congruent sides
- diagonals bisect angles
- diagonals perpendicular
*Square
Hey, look at me!
I have all of the properties of the
parallelogram AND the rectangle AND
the rhombus.
I have it all!
The square is the
most specific
member of the
quadrilateral family.
It has the largest
number of
properties.
Theorems Dealing with
Parallelograms
Topic Index | Geometry Index | Regents Exam Prep Center
*Parallelogram
Definition: A parallelogram is a
quadrilateral with both pairs of opposite
sides parallel.
I have:
- 2 sets of parallel sides
- 2 sets of congruent sides
- opposite angles congruent
- consecutive angles
supplementary
- diagonals bisect each other
- diagonals form 2 congruent
triangles
Using this definition, the remaining properties regarding a parallelogram can be "proven" true and become
theorems.
When GIVEN a parallelogram, the definition and theorems
are stated as ...
A parallelogram is a quadrilateral with both pairs of opposite sides
parallel.
If a quadrilateral is a parallelogram, the 2 pairs of opposite sides are
congruent.
(Proof appears further down the page.)
If a quadrilateral is a parallelogram, the 2 pairs of opposite angles are
congruent.
If a quadrilateral is a parallelogram, the consecutive angles are
supplementary.
If a quadrilateral is a parallelogram, the diagonals bisect each other.
If a quadrilateral is a parallelogram, the diagonals form two
congruent triangles.
When trying to PROVE a parallelogram, the definition and
theorems are stated as ...
(many of these theorems are converses of the previous theorems)
A parallelogram is a quadrilateral with both pairs of opposite sides
parallel.
If both pairs of opposite sides of a quadrilateral are congruent, the
quadrilateral is a parallelogram.
If both pairs of opposite angles of a quadrilateral are congruent, the
quadrilateral is a parallelogram.
If one angle is supplementary to both consecutive angles in a
quadrilateral, the quadrilateral is a parallelogram.
If the diagonals of a quadrilateral bisect each other, the quadrilateral
is a parallelogram.
If ONE PAIR of opposite sides of a quadrilateral are BOTH parallel
and congruent, the quadrilateral is a parallelogram. (Proof appears
further down the page.)
** Be sure to remember this last method, as it may save you time when
working a proof.
Proof of Theorem:
If a quadrilateral is a parallelogram, the 2 pairs of opposite sides are
congruent.
(Remember: when attempting to prove a theorem to be true,
you cannot use the theorem as a reason in your proof.)
STATEMENTS
1
2 Draw segment from A to C
3
4
REASONS
1 Given
2 Two points determine exactly one line.
3 A parallelogram is a quadrilateral with both
pairs of opposite sides parallel.
4 If two parallel lines are cut by a transversal,
the alternate interior angles are congruent.
5 Reflexive property: A quantity is congruent
to itself.
6 ASA: If two angles and the included side of
one triangle are congruent to the
corresponding parts of another triangle, the
triangles are congruent.
7 CPCTC: Corresponding parts of congruent
triangles are congruent.
5
6
7
Proof of Theorem:
If ONE PAIR of opposite sides of a quadrilateral are BOTH parallel
and congruent, the quadrilateral is a parallelogram.
(Remember: when attempting to prove a theorem to be true,
you cannot use the theorem as a reason in your proof.)
STATEMENTS
REASONS
1
1 Given
2 Draw segment from A to C
3
2 Two points determine exactly one line.
3 If two parallel lines are cut by a transversal,
the alternate interior angles are congruent.
4 Reflexive property: A quantity is congruent
to itself.
4
5
6
7
8
5 SAS: If two sides and the included angle of
one triangle are congruent to the
corresponding parts of another triangle, the
triangles are congruent.
6 CPCTC: Corresponding parts of congruent
triangles are congruent.
7 If two lines are cut by a transversal and the
alternate interior angles are congruent, the
lines are parallel.
8 A parallelogram is a quadrilateral with both
pairs of opposite sides parallel.
Theorems Dealing with
Rectangles, Rhombuses,
Squares
Topic Index | Geometry Index | Regents Exam Prep Center
Definition: A rectangle is a
parallelogram with four right angles.
Using the definition, the properties of the rectangle
can be "proven" true and become theorems.
When dealing with a rectangle, the
definition and theorems are stated
as ...
1.
*Rectangle
I have all of the properties
of the parallelogram PLUS
- 4 right angles
- diagonals congruent
A rectangle is a parallelogram with four right
angles.
While the definition states "parallelogram", it is sufficient to
say: "A quadrilateral is a rectangle if and only if it
has four right angles.", since any quadrilateral with four
right angles is a parallelogram.
2.
3.
If a parallelogram has one right angle it is a
rectangle.
A parallelogram is a rectangle if and only if its
diagonals are congruent.
Construction workers use the fact that the diagonals of a rectangle are congruent
(equal) when attempting to build a "square" footing for a building, a patio, a
fenced area, a table top, etc. Workers measure the diagonals. When the
diagonals of the project are equal the building line is said to be square.
Definition: A rhombus is a
parallelogram with four
congruent sides.
Using the definition, the properties of the rhombus
can be "proven" true and become theorems.
When dealing with a rhombus, the
definition and theorems are stated
as ...
*Rhombus
1. A rhombus is a parallelogram with four
congruent sides.
I have all of the properties of While the definition states "parallelogram", it is sufficient
the parallelogram PLUS
to say: "A quadrilateral is a rhombus if and only if it
has four congruent sides.", since any quadrilateral with
- 4 congruent sides
four congruent sides is a parallelogram.
- diagonals bisect angles
- diagonals perpendicular
2. If a parallelogram has two consecutive sides
congruent, it is a rhombus.
3. A parallelogram is a rhombus if and only if each
diagonal bisects a pair of opposite angles.
4. A parallelogram is a rhombus if and only if the
diagonals are perpendicular.
(Proof of theorem appears further down page.)
Definition: A square is a
parallelogram with four congruent
sides and four right
angles.
Using the definition, the properties of the rhombus
can be "proven" true and become theorems.
When dealing with a square, the
definition is stated as ...
A square is a parallelogram with four congruent
sides and four right angles.
This definition may also be stated as
*Square
Hey, look at me!
I have all of the properties of
the parallelogram AND the
A quadrilateral is a square if and only if it is a
rhombus and a rectangle.
rectangle AND the rhombus.
I have it all!
Proof of Theorem:
If a parallelogram is a rhombus, then the diagonals are
perpendicular.
(Remember: when attempting to prove a theorem to be true,
you cannot use the theorem as a reason in your proof.)
STATEMENTS
REASONS
1
1 Given
2 Draw segment from A to C
3
2 Two points determine exactly one line.
3 A rhombus is a parallelogram with four
congruent sides.
4 A rhombus is a parallelogram with four
congruent sides.
4
5
6
7
8
9
5 If a quadrilateral is a parallelogram, the
diagonals bisect each other.
6 A bisector of a segment intersects the
segment at its midpoint.
7 Midpoint of a line segment is the point on
that line segment that divides the segment
two congruent segments.
8 Reflexive Property - A quantity is congruent
to itself.
9 SSS - If three sides of one triangle are
congruent to three sides of a second triangle,
10
11
12
the triangles are congruent.
10 CPCTC - Corresponding parts of congruent
triangles are congruent.
11 If 2 congruent angles form a linear pair, they
are right angles.
12 Perpendicular lines meet to form right
angles.
Theorems Dealing with
Trapezoids
Topic Index | Geometry Index | Regents Exam Prep Center
Definition: A trapezoid is a quadrilateral with exactly one pair of parallel sides.
*Trapezoid
I have only one set of parallel
sides.
[The median of a trapezoid is
parallel to the bases and equal to
one-half the sum of the bases.]
A trapezoid has ONLY ONE set of parallel sides.
When proving a figure is a trapezoid, it is necessary to prove
that two sides are parallel and two sides are NOT parallel.
The median (also called the mid-segment) of a
trapezoid is a segment that connects the
midpoint of one leg to the midpoint of the other
leg.
;
Theorem: The median (or mid-segment) of a
trapezoid is parallel to each base and its length
is one half the sum of the lengths of the bases.
(True for ALL trapezoids.)
Definition: An isosceles trapezoid is a trapezoid with congruent legs.
Theorems:
1. A trapezoid is isosceles if and only if the base
angles are congruent.
2. A trapezoid is isosceles if and only if the
*Isosceles
Trapezoid
I have:
- only one set of parallel sides
- base angles congruent
- legs congruent
- diagonals congruent
- opposite angles supplementary
diagonals are congruent.
3.
If a trapezoid is isosceles, the opposite angles are
supplementary.
Never assume that a trapezoid is isosceles unless you are
given (or can prove) that information.