5.10 Prop. of Rhom., Rect., and Squares
Rhombus Corollary
A quadrilateral is a rhombus if and only if it has
four congruent _______.
sides
A
B
D
C
ABCD is a rhombus if and only if
AB  BC  CD  AD.
5.10 Prop. of Rhom., Rect., and Squares
Rectangle Corollary
A quadrilateral is a rectangle if and only if
it has four _____________.
right angles
A
B
D
C
ABCD is a rectangle if and only if
A, B, C, and D are right angles.
5.10 Prop. of Rhom., Rect., and Squares
Square Corollary
A quadrilateral is a square if and only if it is a
rectangle
________
rhombus and a __________.
A
B
D
C
ABCD is a square if and only if
AB  BC  CD  AD and
A, B, C, and D are right angles.
5.10 Prop. of Rhom., Rect., and Squares
Example 1 Use properties of special quadrilaterals
For any rhombus RSTV, decide whether the statement
is always or sometimes true. Draw a sketch and explain
R
S
your reasoning.
a. S  V
a. By definition, a rhombus is a
parallelogram with four
sides
congruent _______.
V
T
By Theorem 5.19, opposite angles of
congruent So, S  V.
a parallelogram are ___________.
always true.
The statement is _________
5.10 Prop. of Rhom., Rect., and Squares
Example 1 Use properties of special quadrilaterals
For any rhombus RSTV, decide whether the statement
is always or sometimes true. Draw a sketch and explain
your reasoning.
S
R
b. T  V
b. If rhombus RSTV is a
square then all four angles
_______,
are congruent right angles
V
square
So T  V if RSTV is a ________.
squares
Because not all rhombuses are also ___________,
sometimes true.
the statement is ___________
T
5.10 Prop. of Rhom., Rect., and Squares
Example 2 Classify special quadrilaterals
Classify the special quadrilateral.
Explain your reasoning.
The quadrilateral has four congruent _____.
sides
right angle
One of the angles is not a ____________,
square
so the rhombus is not also a ________.
By the Rhombus Corollary, the
quadrilateral is a rhombus
__________.
127o
5.10 Prop. of Rhom., Rect., and Squares
Checkpoint. Complete the following exercises.
1. For any square CDEF, is it always or sometimes
true that CD  DE? Explain your reasoning.
C
D
F
E
By the Square Corollary, all the sides
are congruent.
Therefore, it is always true.
5.10 Prop. of Rhom., Rect., and Squares
Checkpoint. Complete the following exercises.
2. A quadrilateral has four congruent sides and four
congruent angles. Classify the quadrilateral.
4 x  360
x  90
o
C
x
x
x
x
D
o
By the Square Corollary, all the sides
are congruent and all angles are right
angles.
The quadrilateral is a square.
F
E
5.10 Prop. of Rhom., Rect., and Squares
Theorem 5.26
A parallelogram is a rhombus if and only if its
diagonals are _____________.
perpendicular
A
B
D
C
ABCD is a rhombus if and only if
____
AC  ____
BD .
5.10 Prop. of Rhom., Rect., and Squares
Theorem 5.27
A parallelogram is a rhombus if and only if
each diagonal bisects a pair of opposite
sides.
A
B
D
C
ABCD is a rhombus if and only if
BAD
AC bisects  ____
BCD and  ____
ADC .
BD bisects  ____
ABC and  ____
5.10 Prop. of Rhom., Rect., and Squares
Theorem 5.28
A parallelogram is a rectangle if and only if
its diagonals are __________.
congruent
A
B
D
C
ABCD is a rectangle if and only if
____
AC  ____
BD .
5.10 Prop. of Rhom., Rect., and Squares
Example 3 List properties of special parallelograms
F
Sketch rhombus FGHJ.
List everything you know about it.
Solution
G
By definition, you need to draw a
figure with the following properties:
parallelogram J
• The figure is a _______________.
H
sides
• The figure has four congruent _______.
Because FGHJ is a parallelogram, it has these properties:
• opposite sides are ________
parallel and ___________.
congruent
• opposite angles are __________.
congruent Consecutive angles are
supplementary
_________________.
• diagonals ________
bisect each other.
perpendicular
By Theorem 5.26, the diagonals of FGHJ are _______________.
By Theorem 5.27, each diagonals bisect a pair of _____________.
opposite angles
5.10 Prop. of Rhom., Rect., and Squares
Example 4 Solve a real-world problem
Framing You are building a frame
for a painting. The measurements of
the frame are shown at the right.
a. The frame must be a rectangle. Given
the measurements in the figure, can
you assume that it is? Explain.
Solution
No, you cannot. The boards on opposite sides are the same
length, so they form a ______________.
parallelogram
right angles
But you do not know whether the angles are ______________.
5.10 Prop. of Rhom., Rect., and Squares
Example 4 Solve a real-world problem
Framing You are building a frame
for a painting. The measurements of
the frame are shown at the right.
b. You measure the diagonals of the
frame. The diagonals are about 25.6
inches. What can you conclude about
the shape of the frame?
Solution
By Theorem 5.28, the diagonals of a rectangle are congruent
__________.
The diagonals of the frame are congruent
__________, so the frame
rectangle
forms a ______________.
5.10 Prop. of Rhom., Rect., and Squares
Checkpoint. Complete the following exercises.
3. Sketch rectangle WXYZ.
List everything you know about it.
The drawing of WXYZ is
a parallelogram with four
right angles.
W
X
Z
Y
It has these characteristics:
• opposite sides are parallel and congruent.
• opposite angles are congruent and consecutive angles
are supplementary.
• diagonals bisect each other and are congruent.
5.10 Prop. of Rhom., Rect., and Squares
Checkpoint. Complete the following exercises.
4. Suppose the diagonals of the frame
in Example 4 are not congruent.
Could the frame still be a rectangle?
By Theorem 5.28, a rectangle must have congruent diagonals.
5.10 Prop. of Rhom., Rect., and Squares
Pg. 331, 5.10 #2-24
even, 25