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6.4 Rhombi, Rectangles,
and Squares
California State Standards for Geometry
4: Prove basic theorems involving congruence & similarity.
7: Prove & use theorems involving parallel lines and
properties of quadrilaterals.
12: Find & use side and angle measures of triangles
and polygons
17: Prove theorems using coordinate geometry
definitions
Rhombus
A parallelogram with all sides congruent
Rectangle
A parallelogram with all angles congruent
(4 right angles)
Square
A parallelogram with all sides congruent
and all angles congruent
created by chris
markstrum 2005
created by chris
markstrum 2005
Quadrilaterals
parallelograms
squares
theorems about rhombi,
rectangles, and squares
Rhombus Corollary
A quadrilateral is a rhombus if and only if
it has four congruent sides.
Rectangle Corollary
A quadrilateral is a rectangle if and only if
it has four right angles.
Square Corollary
A quadrilateral is a square if and only if
it is both a rectangle and a rhombus.
created by chris
markstrum 2005
created by chris
markstrum 2005
theorem
A parallelogram is a rhombus if and only if
its diagonals are perpendicular.
If in
ABCD
B
then it is
a rhombus
AC  BD
A
C
D
theorem
created by chris
markstrum 2005
A parallelogram is a rhombus if and only if
each diagonal bisects a pair of opposite angles.
If in
ABCD
B
then it is
a rhombus
AC bisects
A and C
and
BD bisects
B and D
A
C
D
theorem
created by chris
markstrum 2005
A parallelogram is a rectangle if and only if
its diagonals are congruent.
If in
ABCD
AC  BD
B
A
AC  BD
then it is
a rectangle
D
C
Given: AC  BD
Prove:
ABCD is a rectangle
1.
ABCD, AC  BD
A
D
1. Given
2. AB  CD, AD  BC
2.
ACD  BDC
4. ADC  BCD
5. DAB  BCD,
ADC  CBA
6. A  B  C  D
3.
7. ABCD is a rectangle
7. Def rectangle
3.
  opp. sides
SSS 
4. C.P.C.T.C.
5.
  opp.  ' s
6. Transitive Prop
B
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markstrum 2005
C
Given :
Prove :
AC  BD
ABCD is a rectangle
Statement
1.
2.
ABCD , AC  BD
AB  CD, AD  BC
created by chris
markstrum 2005
A
B
D
C
Reason
1.
2.
Given
  opp. sides
3.
ACD  BDC
3.
SSS 
4.
ADC  BCD
4.
C.P.C.T.C.
5.
DAB  BCD,
ADC  CBA
5.
6.
A  B  C  D
6.
Transitive
7.
ABCD is a rectangle
7.
Def rectangle
  opp.  ' s
Example
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markstrum 2005
A rectangle is a square.
Sometimes, Always, or Never?
Quadrilaterals
parallelograms
squares
Example
created by chris
markstrum 2005
A square is a rhombus
Sometimes, Always, or Never?
Quadrilaterals
parallelograms
squares
Example
created by chris
markstrum 2005
A rhombus is a rectangle
Sometimes, Always, or Never?
Quadrilaterals
parallelograms
squares
Example
created by chris
markstrum 2005
A parallelogram is a rectangle
Sometimes, Always, or Never?
Quadrilaterals
parallelograms
squares
Example
created by chris
markstrum 2005
rectangle EFGH, EG = 8z – 16
Find EK and GK
E
K
EK  12 EG
EK  12 8z 16
EK  4z  8
GK  EK
GK  4z  8
F
H
The diagonals of a rectangle
bisect each other.
G
Example
Rectangle ABCD
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markstrum 2005
mB  8x  26
Find the value of x
A
B
D
C
By definition of a rectangle,
B is a right angle.
mB  90
8x  26  90
x 8
Example
Rectangle MATH
Perimeter MAT  30
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markstrum 2005
M
MA  AT  4 x  7
MT  2 x  1
H
Find AH and the perimeter of rect. MATH
( MA  AT )  MT  30
(4 x  7)  2 x  1  30
x6
A
T
Example
created by chris
markstrum 2005
Rectangle MATH
Perimeter MAT  30
M
MA  AT  4 x  7
MT  2 x  1
H
Find AH and the perimeter of rect. MATH
x6
AH  MT  2 6 1
AH  13
A
T
Example
Rectangle MATH
Perimeter MAT  30
created by chris
markstrum 2005
M
MA  AT  4 x  7
MT  2 x  1
H
Find AH and the perimeter of rect. MATH
x6
perimeter of rect. MATH = 2( MA  AT )
 2(4 x  7)  2(4 6  7)
 34
A
T
Summary



What are the three ways to show that a
parallelogram is a rhombus
What are the two ways to show that a
parallelogram is a rectangle?
How do you show that a parallelogram is a
square?
created by chris
markstrum 2005