Download 8.4 Properties of Rhombuses, Rectangles and Squares

8.4 Properties of Rhombuses,
Rectangles, and Squares
Hubarth
Geometry
A rhombus is a parallelogram with four congruent sides.
A rectangle is a parallelogram with four right angles.
A square is a parallelogram with four congruent sides and four right angles.
Ex 1. Use Properties of Special Parallelograms
In the diagram, ABCD is a rectangle .
A
B
5
a. Find AD and AB
b. Find mA, mB, mC and mD.
Solution
a. AD=BC=5 and AB=DC=8
b. A=B=C=D=90
D
8
C
Corollaries
Rhombus Corollary
Words
Symbols
If a quadrilateral has four congruent sides,
then it is a rhombus.
B
A
If AB  BC  CD  AD, then ABCD is
a rhombus
C
D
Rectangle Corollary
Words
If a quadrilateral has four right angles,
then it is a rectangle.
Symbols If mA=mB=mC=mD=90,
then ABCD is a rectangle.
Square Corollary
Words
If a quadrilateral has four congruent sides
and four right angles, then it is a square.
Symbols If AB  BC  CD  AD and the
mA=mB=mC=mD=90,
then ABCD is a square.
A
D
B
C
Ex 2. Identify Special Quadrilaterals
Use the information in the diagram to name the special quadrilateral.
3
5
Solution
The quadrilateral has four right angles, but does not have four congruent sides.
So, the quadrilateral is a rectangle.
Theorem 8.11
Words
Symbols
The diagonals of a rhombus are perpendicular
In rhombus ABCD, AC  BD
A
D
B
C
Ex 3. Use Diagonals of a Rhombus
A
ABCD is a rhombus.
Find the value of x.
B
60
X
E
D
C
Solution
Because ABCD is a rhombus. The diagonals are perpendicular.
AEB is a right triangle.
So, 90+60+x=180
150+x=180
-150 -150
x=30
Theorem 8.12
A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles
Theorem 8.13
Words
Symbols
The diagonals of a rectangle are congruent.
In rectangle ABCD, AC  BD
A
B
D
C
Ex 4. Use Diagonals of a Rectangle
a. You nail four pieces of wood together to build a four-sided frame, as shown.
What is the shape of the frame?
b. The diagonals measure 7ft. 4 in. and 7 ft 2 in. Is the frame a rectangle?
4
6
6
4
Solution
a. The frame is a parallelogram because both pairs of opposite sides are congruent.
b. The frame is not a rectangle because the diagonals are not congruent.
Practice
S
7
1. In the diagram, RSTU is a square. Find the UT and RS
and mR and mT
UT  RS  7
R
T
mR  mT  90
U
2. Use the information in the diagram to name the quadrilateral.
14
Square
14
M
3. JKLM is a rhombus. Find the value of x.
x= 40
14
J
T
L 50 x
K
4. ABCD is a rectangle. Find the value of x.
A
10
B
x=10
x
D
14
C