Transcript Properties of Parallellograms


By definition, Opposite Sides are Parallel
2
1
3
Opposite Angles are Congruent
 Consecutive Angles are Supplementary

2
1
3
Angles 1 & 2 are
Same Side Interior
So are Supplementary
Angles 2 & 3 are also
Same Side Interior
So are Supplementary
Since angles 1 & 3 are
Both supplementary to
Angle 2, they must be
Congruent

Opposite Sides are Congruent
2
1
4
Angles 1 & 3 are
Alternate Interior
So are Congruent
3
Angles 2 & 4 are also
Alternate Interior
So are Congruent
The segment BD is
Congruent to itself by the
Reflexive Property
Triangle DAB and
Triangle BCD are
Congruent by ASA
AD ≅BC and AB ≅ DC
By CPCTC

Diagonals Bisect each other
Angles 1 & 3 are
Alternate Interior
So are Congruent
2
1
3
E
4
Angles 2 & 4 are
Vertical Angles
So are Congruent
AD ≅ BC because
Opposite Sides ≅
Triangle DAE and
Triangle BCE are
Congruent by AAS
AE ≅CE and DE ≅ BE
By CPCTC
AC and DB bisect
Each other by definition
By definition a rectangle is a quadrilateral
with all 4 right angles.
Could also be defined as a parallelogram
with at least 1 right angle.
Diagonals of a rectangle are congruent
B
2
DC is congruent
To itself by the
Reflexive Property
1
D
Angles 1 & 2 are
Alternate Interior
So are congruent
C
Angles ADC & BCD
are Right Angles
So are congruent
Diagonals of a rectangle are congruent
B
2
Angles 1 & 2 are
Alternate Interior
So are congruent
DC is congruent
To itself by the
Reflexive Property
1
D
C
Angles ADC & BCD
are Right Angles
So are congruent
Triangles ADC & BCD
are congruent by
AAS
AD ≅ BC by CPCTC
By definition a rhombus is a quadrilateral
with all 4 sides congruent.
Could also be defined as a parallelogram
with congruent consecutive sides.
Diagonals are Perpendicular.
Any point that is equidistant from the endpoints of
a segment is on the perpendicular bisector.
Diagonals Bisect Opposite Angles.
A
D
C
B
Angle ACB and ACD
are right angles so
are congruent
AD ≅ AB by definition of Rhombus
AC ≅ AC by Reflexive property
ΔACD ≅ Δ ACB by HL
∠DAC ≅ ∠BAC by CPCTC
Is both a Rectangle and a Rhombus,
so has all the properties of both.
Diagonals are Congruent
and Perpendicular
and bisect opposite angles
Creating isosceles right Δs
Property
Parallelogram Rectangle
Rhombus
Square
Opposite sides ‖
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x
x
x
Opposite sides ≅
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x
x
x
Opposite Angles ≅
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x
x
x
Consecutive Angles
Supplementary
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x
x
x
Diagonals Bisect
Each other
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x
x
x
4 Right Angles
x
x
Diagonals ≅
x
x
Diagonals ⊥
x
x
Diagonals Bisect
Opposite Angles
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x