Advanced Math I
Unit 3 Lesson 2 Properties, Definitions and Theorems
E
Parallelogram: a quadrilateral is a parallelogram if and only if both pairs of
opposite sides are parallel.
F
Ex: If EF // HG and EH // FG then EFGH is a paralle log ram.
If EFGH is a paralle log ram then EH // FG and EF // HG.
H
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From the above definition of a parallelogram we can prove the following:
(These are minimal requirements you can use to prove that a quadrilateral is in fact a parallelogram,
besides the definition.)
 If a quadrilateral has two pairs of opposite sides congruent,
then it’s a parallelogram.
 If a quadrilateral has two pairs of opposite angles congruent,
then it’s a parallelogram.
 If a quadrilateral has one pair of opposite sides congruent
and parallel, then it’s a parallelogram.
 If a quadrilateral has diagonals that bisect each other,
then it’s a parallelogram.
G
B
B
A
C
C
A
D
D
Properties of a Parallelogram:
 Opposite sides are congruent. AB  DC and BC  AD .
 Opposite angles are congruent. A  C and B  D.
 Consecutive angles are supplementary.
A  B 180, B  C 180, C  D 180, and D A 180.
 Diagonals bisect each other. BD bisects AC and AC bisects BD.
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 Rectangle: A quadrilateral is a rectangle if and only if it is a parallelogram
with one right angle.
Because a rectangle is a parallelogram it has all of the properties that
a parallelogram has:
 Opposite sides are congruent and parallel.
 Opposite angles are congruent.
 Consecutive angles are supplementary.
 Diagonals bisect each other.
Additionally, a rectangle has the following properties:
 Diagonals are congruent. DF  GE
 Consecutive angles are congruent. D  E  F  G
D
E
G
F
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Rhombus: a parallelogram witha pair of congruent adjacent sides.
Because a rhombus is a parallelogram it has all of the properties that a parallelogram has:
 Opposite sides are congruent and parallel.
 Opposite angles are congruent.
 Consecutive angles are supplementary.
 Diagonals bisect each other.
X
Additionally, a rhombus has the following properties:
 Diagonals are perpendicular.
WVX XVY YVZ ZVW  90
 The diagonals bisect the angles.
WXV  YXV, XYV  ZYV, YZV  WZV, and ZWV  XWV.
W
V
Y
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Z
Square: A parallelogram with one right angle and one pair of adjacent sides congruent.
A square is a rectangle and a rhombus and therefore has all of the following properties:
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

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


All sides are congruent. AB  BC  CD  AD
Opposite sides are parallel. AB PCD and BC PAD
All angles are congruent. ABC  BCD  CDA  DAB
Consecutive angles are supplementary. ABC  BCD  180
Diagonals bisect each other. AC bisects BD and BD bisects AC
Diagonals are perpendicular. AC  BD
The diagonals bisect the angles. BD bisects ABC & ADC
Diagonals are congruent. BD  AC
A
B
E
D
C
A
Kite: A quadrilateral with two sets of distinct consecutive sides congruent.
B
A kite has all of the following properties:
 The shorter diagonal is bisected by the longer diagonal.
AD bisects BC
 The angles that are intercepted by the shorter diagonal are congruent.
ABD  ACD
 The longer diagonal bisects the angles it intercepts.
AD bisects BAC & BDC
 The diagonals are perpendicular.
AD  BC
C
D
A
B
E
D
C
B
A
Trapezoid: A quadrilateral, which has only one set of opposite sites parallel.
A trapezoid has the following property:
 Consecutive angles on different bases are supplementary.
B  C  180 and A  D  180
C
D
X
Y
Isosceles Trapezoid: A trapezoid whose nonparallel sides are congruent.
Z
W
Additional properties of an isosceles trapezoid are:
 Diagonals are congruent. XZ  WY
 Angles on the same base are congruent.
WXY  ZYX & XWZ  YZW
X
Y
Z
W
Midpoint Connector Theorem for Quadrilaterals: If the midpoints of consecutive sides of any
quadrilateral are connected, the resulting quadrilateral is a parallelogram.
Ex: If E, F, G, and H are midpoints then EFGH is a parallelogram.
B
E
F
A
C
H
G
D