# 6.4: Squares and Rhombi

```2.3d:Quadrilaterals
CCSS:
- Squares and Rhombi
G-CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are
congruent, opposite angles are congruent, the diagonals of a parallelogram bisect
each other, and conversely, rectangles are parallelograms with congruent diagonals.
G-GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example,
prove or disprove that a figure defined by four given points in the coordinate plane is
a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the
origin and containing the point (0, 2).
GSE’s
M(G&amp;M)–10–2 Makes and defends conjectures, constructs
geometric arguments, uses geometric properties, or uses
theorems to solve problems involving polygons
Rhombi
A parallelogram with 4 congruent sides
(tilted square)
Since the rhombus is a more specific
parallelogram, all the properties of the
parallelogram can be found in it.
POLYGONS
1
Parallelograms
3)
4)
RECTANGLES
RHOMBI
Has ALL the properties of the
shapes above it
Rhombi Properties
In addition to the parallelogram properties and the
4 congruent sides, its also has 3 other properties
1)The diagonals of a rhombus are perpendicular
(Theorem 6-11)
2) If the diagonals of a parallelogram are
perpendicular, then the parallelogram is a
rhombus
(Theorem 6-12) (the converse of 6-11 which proves a parallelogram is a
rhombus)
How could this theorem help us on the coordinate plane?
If we have a parallelogram, we can use the slopes of the diagonals to
determine they are perpendicular, therefore telling us it is a rhombus.
Rhombi Properties (Con’t)
3) Each diagonal of a rhombus bisects a pair
of opposite angles.
* And opposite angles
are congruent since it
has the properties of
parallelograms
Square
A Square is both a Rectangle and a
Rhombus
1) Has 4 right angles like a rectangle
2) Has 4 congruent sides like a rhombus
POLYGONS
Parallelograms
RECTANGLES
RHOMBI
5)
Square
Has ALL the properties of the
shapes above it
Example
PTRE is a square, solve for x
5x+10=45
5x = 35
x=7
Example
Determine whether the parallelogram ABDC
is a rhombus, rectangle, or square if:
All work must be justified mathematically
A (-4,3)
B (-2,3)
ANS: Square
C (-4, 1)
D (-2,1)
Example 2
Determine whether parallelogram WXYZ is a
rhombus, rectangle, or square.
All work must be justified mathematically
W (1,10)
X (-4,0)
ANS: Square
Y (7,2)
Z (12,12)
Kyle is building a barn for his horse. He
measures the diagonals of the door
opening to make sure that they bisect
each other and they are congruent.
How does he know that the corners are
right angles?
Answer: We know that AC  BD . A parallelogram with
congruent diagonals is a rectangle. Therefore,
o
the corners are 90 angles.
EXAMPLE
Use rhombus LMNP to find the value of y if
Why
y 2  54  90
y 2  144
y 2  144
y  12
N
Venn Diagram
Place the
following
polygons
according to
the Venn
Diagram: