Rhombi and Squares in Geometry
6.5
Rhombi and Squares
Then/Now
You determined whether quadrilaterals were
parallelograms and/or rectangles.
•
Recognize and apply the properties of rhombi
and squares.
•
Determine whether quadrilaterals are
rectangles, rhombi, or squares.
Vocabulary
•
Rhombus:
a parallelogram with all four sides
congruent
Vocabulary
•
Square:
a parallelogram with four congruent sides
and four right angles
Vocabulary
Theorem 6.15:
Diagonals of a Rhombus #1
If a parallelogram is a rhombus, then its
diagonals are perpendicular.
Vocabulary
Theorem 6.16:
Diagonals of a Rhombus #2
If a parallelogram is a rhombus, then
each
diagonal bisects a pair of opposite angles.
Vocabulary
Theorem 6.17:
Condition #1 for a Rhombus
If the diagonals of a parallelogram are
perpendicular, then the parallelogram is a
rhombus.
*Converse of 6.15
Vocabulary
Theorem 6.18:
Condition #2 for a Rhombus
If one diagonal of a parallelogram bisects a
pair of opposite angles, then the parallelogram is a
rhombus.
*Converse of 6.16
Vocabulary
Theorem 6.19:
Condition #3 for a Rhombus
If one pair of consecutive sides of a
parallelogram are congruent, then the
parallelogram is a rhombus.
Vocabulary
Theorem 6.20:
Square Conditions
If a quadrilateral is both a rectangle and a
rhombus, then it is a square.
Example 1A
Use Properties of a Rhombus
A.
The diagonals of rhombus
WXYZ
intersect at
V
.
If
m
WZX
= 39.5, find
m
ZYX
.
Answer:
m
ZYX
= 101
Example 1B
Use Properties of a Rhombus
B. ALGEBRA
The diagonals of rhombus
WXYZ
intersect at
V
.
If
WX
= 8
x
– 5 and
WZ
= 6
x
+ 3, find
x
.
Answer:
x
= 4
Example 1A
A.
m
CDB
= 126
B.
m
CDB
= 63
C.
m
CDB
= 54
D.
m
CDB
= 27
A.
ABCD
is a rhombus. Find
m
CDB
if
m
ABC
= 126.
Example 1B
A.
x
= 1
B.
x
= 3
C.
x
= 4
D.
x
= 6
B.
ABCD
is a rhombus. If
BC
= 4
x
– 5 and
CD
= 2
x
+ 7,
find
x
.
Rectangle
Rhombi
Example 2
Is there enough information given to prove that
ABCD
is a
rhombus?
Proofs Using Properties of Rhombi and Squares
A.
Yes, if one pair of consecutive sides of a
parallelogram are congruent, the
parallelogram is a rhombus.
B.
No, you need more information
Example 3
A.
The diagonal bisects a pair of opposite
angles.
B.
The diagonals bisect each other.
C.
The diagonals are perpendicular.
D.
The diagonals are congruent.
Sachin has a shape he knows to be a parallelogram and all
four sides are congruent. Which information does he need
to know to determine whether it is also a square?
Explore the properties and theorems related to rhombi, squares, and parallelograms. Learn how to identify rhombi, squares, and rectangles based on their properties and conditions. Enhance your knowledge of diagonals, angles, and congruency in quadrilaterals through examples and vocabulary explanations.
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Presentation Transcript
6.5 Rhombi and Squares
Then/Now You determined whether quadrilaterals were parallelograms and/or rectangles. Recognize and apply the properties of rhombi and squares. Determine whether quadrilaterals are rectangles, rhombi, or squares.
Vocabulary Rhombus: a parallelogram with all four sides congruent
Vocabulary Square: a parallelogram with four congruent sides and four right angles
Vocabulary Diagonals of a Rhombus #1 Theorem 6.15: If a parallelogram is a rhombus, then its diagonals are perpendicular.
Vocabulary Diagonals of a Rhombus #2 Theorem 6.16: diagonal bisects a pair of opposite angles. If a parallelogram is a rhombus, then each
Vocabulary Condition #1 for a Rhombus Theorem 6.17: If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. *Converse of 6.15
Vocabulary Condition #2 for a Rhombus Theorem 6.18: pair of opposite angles, then the parallelogram is a rhombus. If one diagonal of a parallelogram bisects a *Converse of 6.16
Vocabulary Condition #3 for a Rhombus Theorem 6.19: If one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus.
Vocabulary Square Conditions Theorem 6.20: If a quadrilateral is both a rectangle and a rhombus, then it is a square.
Example 1A Use Properties of a Rhombus A. The diagonals of rhombus WXYZ intersect at V. If m WZX = 39.5, find m ZYX. Answer:m ZYX = 101
Example 1B Use Properties of a Rhombus B. ALGEBRA The diagonals of rhombus WXYZ intersect at V. If WX = 8x 5 and WZ = 6x + 3, find x. Answer:x = 4
Example 1A A. ABCD is a rhombus. Find m CDB if m ABC = 126. A. m CDB = 126 m CDB = 63 B. m CDB = 54 C. D. m CDB = 27
Example 1B B. ABCD is a rhombus. If BC = 4x 5 and CD = 2x + 7, find x. A. x = 1 B. x = 3 C. x = 4 D. x = 6
Rectangle Rhombi
Example 2 Proofs Using Properties of Rhombi and Squares Is there enough information given to prove that ABCD is a rhombus? Given: ABCD is a parallelogram. AD DC Prove: ABCD is a rhombus A. Yes, if one pair of consecutive sides of a parallelogram are congruent, the parallelogram is a rhombus. B. No, you need more information
Example 3 Sachin has a shape he knows to be a parallelogram and all four sides are congruent. Which information does he need to know to determine whether it is also a square? A. The diagonal bisects a pair of opposite angles. B. The diagonals bisect each other. C. The diagonals are perpendicular. D. The diagonals are congruent.
