Geometry
Quadrilaterals Proofs
Name: ______________________________
How do we know all those properties about parallelograms and other quadrilaterals are true? Where do
they come from? Well, here are some proofs that will help us “discover” those properties. Most of the
proofs involve using congruent triangles and angles formed by parallel lines.
Property: Opposite sides of a parallelogram are congruent.
Given: ABCD is a parallelogram.
Prove: AB  CD ; BC  DA
Statements
ABCD is a parallelogram.
Justifications
AB || DC ; BC || AD
1  3 ; 2  4
AC  AC
ABC  CDA
AB  CD ; BC  DA
*Definition of Congruent Triangles (CPCTC:
is used once you have proven 2 triangles congruent. It allows you to say that ANY of the non-mentioned
corresponding parts in both triangles are congruent. Because if the 2 triangles have 3 corresponding parts
congruent, they have all corresponding parts congruent.
Property: Opposite angles of a parallelogram are congruent.
Given: Parallelogram MNPQ
Prove: M  P
Statements
MNPQ is a parallelogram
Justifications
mM  mN  180
mN  mP  180
Transitive Property of equality
Subtraction Property of equality
The exact same steps would follow to show that N  Q .
Property: The diagonals of a parallelogram bisect each other.
Given: Parallelogram ABCD
Prove: AC bisects BD
Statements
Justifications
ABCD is a parallelogram
AB || CD
ABD  _______
BAC  _______
Alternate Interior Angles Theorem
AB  DC
ABE  ________
AE  EC
DE  EB
Definition of Segment Bisector
Property: Each diagonal of a rhombus bisects two angles of the rhombus.
Given: Rhombus ABCD
Prove: AC bisects BAD and BCD
Statements
ABCD is a rhombus
Justifications
AB  BC  CD  AD
Reflexive Property
ADC  ABC
1  2
_____  ______
Definition of Congruent Triangles (CPCTC)
Definition of Bisect
Property: The diagonals of a rhombus are perpendicular.
Give: Rhombus ABCD
Prove: AC  BD
Statements
ABCD is a rhombus
Justifications
AB  BC  CD  AD
AE  EC
BE  ED
ABE  CBE
BEA   _______
Definition of Congruent Triangles (CPCTC)
mBEA  mBEC  180
mBEA  mBEA  180
2(mBEA)  180
mBEA  90
AC  BD
Property: The diagonals of a rectangle are congruent.
Given: Rectangle ABCD
Prove: AC  BD
Statements
Justifications
ABCD is a rectangle
AB  CD
BC  CD
BAD  90
CDA  90
BAD  CDA
BAD  CDA
AC  BD
Opposite sides of a parallelogram are ____________
Definition of a ________________________