# First, let’s review how to prove a parallelogram. B C

```First, let’s review how to prove a parallelogram.
Given:
AB CD
ABD
CDB
Prove:
ABCD is a
Statements
1. AB CD
2.
ABD
CDB
3. AB || CD
4. ABCD is a
B
C
A
D
Reasons
1. Given
2. Given
3. If alternate interior angles are , then
the lines are ||
4. If opp. sides of a quadrilateral are both
|| and , then the figure is a
parallelogram.
There are several quadrilaterals that we will need
to prove are specific shapes. They include:
 Rectangle
 Kite
 Rhombus
 Square
 Isosceles Trapezoid
For several of these, we will first need to prove
the figure is a parallelogram.
Let’s begin.
Proving Rectangles
When proving rectangles you have 3 options.
1st – If you can prove a quadrilateral has 4 right
angles, then the quadrilateral is a rectangle.
2nd – If you can 1st prove the quadrilateral is a
parallelogram, then prove at least 1 right angle,
then the quadrilateral is a rectangle.
3rd – If you can 1st prove the quadrilateral is a
parallelogram, then prove the diagonals are
congruent, then the quadrilateral is a rectangle.
Proving Kites
When proving kites you have 2 options.
1st - If 2 disjoint pairs of consecutive sides are ,
then the figure is a kite.
2nd - If 1 of the diagonals is the bisector of the
other diagonal, then the figure is a kite.
(Show that 1 diagonal is to the other AND that the same
diagonal bisects the other)
Proving Rhombuses
When proving rhombuses you have 3 options.
1st - If the diagonals are bisectors of each other, then the
figure is a rhombus.
2nd - There must be a step in the proof that says the figure
is a PARALLELOGRAM AND If a parallelogram
contains a pair of consecutive sides that are , then the
figure is a rhombus
3rd - There must be a step in the proof that says the figure
is a PARALLELOGRAM AND If either diagonal of a
parallelogram bisects 2 ’s of the parallelogram, then
the figure is a rhombus.
Proving Squares
When proving squares you have 1 option.
1st - If a quadrilateral is BOTH a rectangle and a rhombus,
then the figure is a square.
(So you MUST have a step in the proof that shows the figure
is a rectangle and another step that show the figure is a
rhombus.)
Proving Isosceles Trapezoids
When proving Isos. Trapezoids you have 3 options.
1st - If the non-parallel sides of a trapezoid are , then the
figure is an isosceles trapezoid.
2nd - If the lower or upper base ’s of a trapezoid are ,
then the figure is an isosceles trapezoid.
3rd - If the diagonals of a trapezoid are , then the figure is
an isosceles trapezoid.
In review to prove:
Rectangles ( 3 methods)
If all 4 ’s are right ’s, then the figure is a rectangle.
Prove a parallelogram and
1.
2.

If a parallelogram contains at least 1 right
rectangle.
, or if the diagonals are , then the figure is a
Kites (2 methods)
If 2 disjoint pairs of consecutive sides are , then the figure is a kite.
If 1 of the diagonals is the bisector of the other diagonal, then the figure is a kite.
1.
2.
Rhombus (3 methods)
If the diagonals are bisectors of each other, then the figure is a rhombus.
Prove a parallelogram and
1.
2.

If a parallelogram contains a pair of consecutive sides that are , or if either diagonal of a
parallelogram bisects 2 ’s of the parallelogram, then the figure is a rhombus.
Squares (1 method)
1.
If a quadrilateral is BOTH a rectangle and a rhombus, then the figure is a square.
Isosceles Trapezoids (3 methods)
1.
2.
3.
If the non-parallel sides of a trapezoid are , then the figure is an isosceles trapezoid.
If the lower or upper base ’s of a trapezoid are , then the figure is an isosceles
trapezoid.
If the diagonals of a trapezoid are , then the figure is an isosceles trapezoid.
First, let’s review how to prove a parallelogram.
Given:
GJMO is a
OH JK
MK is an alt. of ▲ MKJ
Prove:
OHKM is a rectangle
O
G
Statements
M
J
H
K
Reasons
1.
2.
3.
4.
5.
6.
GJMO is a
OM || GK
OH GK
MK is an alt. of ▲MKJ
MK GK
OH || MK
1.
2.
3.
4.
5.
6.
7.
OHKM is a
7.
8.
9.
OHK is a right
OHKM is a rectangle
8.
9.
Given
Opp. Sides of a
are ||
Given
Given
Def. of altitude
If two coplanar lines are to a 3rd line, then
they are ||.
If both pairs of opp. sides of a quad. are ||,
then it is a
Def. of
If a
contains at least one right , it is a
rectangle.
```