Download Geometry Notes Name: Proofs of Quadrilateral Properties

Geometry Notes
Proofs of Quadrilateral Properties
Name: _________________
Definitions:
 A figure is a Parallelogram, IFF it is a quadrilateral with two sets of opposite, parallel sides.
 A figure is a Rectangle IFF it is a quadrilateral with four right angles.
 A figure is a Rhombus IFF it is a quadrilateral with four congruent sides.
 A figure is a Square IFF it is a quadrilateral with four congruent sides & four congruent angles.
 A figure is a Trapezoid IFF it is a quadrilateral with exactly 1 pair of parallel lines.
 A figure is an Isosceles Trapezoid IFF it is a Trapezoid and the non parallel sides are congruent.
Skill: Write a proof for each of the following Quadrilateral Properties. Begin by
translating the “If-Then” statement into a Given and Prove statement that applies to the
picture. You should be able to do them as either Flow Proofs, or as 2 column proofs.
2) If Rectangle, then Parallelogram.
1) If Parallelogram, then Opposite Angles Are
Congruent.
Given:
Given:
B
C
ABCD is Rectangle.
B
C
ABCD is Parallelogram.
A
ABCD is
parallelogram
Prove:
Prove: ABC@CDA
D
A
BAC & DCA are Alt Interior Angles
BCA & DAC are Alt Interior Angles
Given
AB / / DC & BC / / DA
D
ABCD is
rectangle
mA  90 ; mB 90 ;
Given
mC  90 ; mD  90


Definition of Alt.Inter.Angles
A, B , C & D
AB // DC & BC // DA
Definition of Parallelogram
AC is transversal
Def. of transversal
AC  AC
BAC  DCA
BCA  DAC

are right angles

Definition Right Angle
mA  mB 180 ;

Definition of Rectangle
mC  mD  180

Addition Prop =
If parallel lines then Alt Int s @
A & BC & D are supplementar
 ABC CDA
ASA
Reflexive Prop of @
Definition of Supplementary
A & BC & D are Same Side Interior Angles
Definition of Same Side Int. Angles
ABC@CDA
CPCTC
AB / / DC & BC / / DA
ABCD is
parallelogra
If SSI angles supplementary,
Definition of Parallelogram
then lines are parallel. QED
QED
3) If Parallelogram, then Opposite Sides are Congruent.
4) If Parallelogram, then Consecutive Angles are Supplementary.
5) If Parallelogram, then Diagonals are Bisected.
6) If Rhombus, then Parallelogram.
7) If Rectangle, then diagonals are congruent.
8) If Rhombus, then diagonals are Perpendicular.
9) If Kite, then diagonals are Perpendicular.
10) If Isosceles Trapezoid, then Diagonals are congruent.
Geometry Quadrilateral Proofs
Worksheet
Name: ____________________
Instructions: Fill in the missing information.
Definitions:
A quadrilateral is a Parallelogram IFF it has two sets of parallel sides.
A quadrilateral is a Rhombus IFF it has four congruent sides.
A quadrilateral is a Rectangle IFF it has four _______ _______.
A quadrilateral is a Square IFF it has ______________ & _____________.
1) B
C If Parallelogram, then Opposite Angles Are Congruent.
A
D
Given:
ABCD is Parallelogram.
Prove: ABC@CDA
1)
2) AB // DC & BC // DA
3) _______&______ are Alt Interior Angles
_______&______ are Alt Interior Angles
4) AC is a transversal
5) BCA  DAC
6)  ______ @  _______
7)
8)  ABC CDA
9) ABC@CDA
1) Given
2) Definition of ___________________
3) Definition of ___________________
4) Definition of Transversal
5)
6)
7) Reflexive prop @
8)
9)
QED
2)
Use the above proof as a guide, and now prove:
If Parallelogram, then Opposite Sides Are Congruent.
B
Given:
ABCD is Parallelogram.
C
Prove: AB  CD
A
& BC  DA
D
3)
1)
1)
2)
2)
3)
3)
4)
4)
5)
5)
6)
6)
7)
7)
8)
8)
9)
9)
QED
Could you prove If Parallelogram, then Opposite Angles Are Congruent much faster, if you already
had the theorem: If Parallelogram, then Opposite Sides Congruent? Do it in 5 steps below.
B
1)
ABCD is Parallelogram.
1)
2)
2)
3)
3)
4)
4)
5) ABC@CDA
5)
Given
A
D
C
4)
Prove: If Parallelogram, then Diagonals are Bisected.
B
Given:
ABCD is Parallelogram with
diagonals/transversals AC and BD
Prove: BX @ XD & AX @ XC
C
X
A
D
1)
1) Given
2)
2) Definition of Parallelogram
3) XAB & XCD are ________________
3) Definition of Alternate Interior Angles
4)
4)
5) AXB & CXD are ________________
5)
6)
6)
7) AB  _____
7) If parallelogram, then opp sides @
8) ABX@__ __ __
8)
9)
9)
QED
5) Prove: If, in quadrilateral, two sets of Sides Are Congruent, then parallelogram.
B
C
Given: AB  CD
A
D
Prove: ABCD is Parallelogram.
1)
1)
2)
2) Reflexive Prop @
3)  ABC @  ___ ___ ___
Given
3)
4) BCA @__ __ __ & DCA @ __ __ __
4)
5) __ __ __ & DAC are AIA (for BC & DA)
5) Definition of AIA
6) __ __ __ & BAC are AIA (for AB & CD)
7)
8)
9)
& BC  DA
6)
7)
8)
9) Definition of Parallelogram
QED
6) Prove: If one pair of opposite sides are both congruent and parallel, then parallelogram.
B
C
Given: BC // DA&BC  DA
A
D
Prove: ABCD is Parallelogram.
QED
Given: ABCD is Rhombus.
B
C
Prove: ABCD is Parallelogram
7) Prove: If Rhombus, then Parallelogram.
A
1)
2) AB  CD
D
1)
2)
& BC  DA
3)
3) Reflexive property of @
4)
4)
5) BAC &______ are Alt Interior Angles
5)
6)
6)
7) AC is transversal to lines AB & CD.
7)
8)
8)
9)
9) Definition of Parallelogram
QED