Download Reteach 6.4

Name ________________________________________ Date ___________________ Class __________________
LESSON
6-4
Reteach
Properties of Special Parallelograms
A rectangle is a quadrilateral with four right angles. A rectangle has the
following properties.
Properties of Rectangles
If a quadrilateral is a rectangle, then it
is a parallelogram.
If a parallelogram is a rectangle, then
its diagonals are congruent.
Since a rectangle is a parallelogram, a rectangle also has all the properties of parallelograms.
A rhombus is a quadrilateral with four congruent sides. A rhombus has the following
properties.
Properties of Rhombuses
If a quadrilateral is a
rhombus, then it is a
parallelogram.
If a parallelogram is a
rhombus, then its diagonals
are perpendicular.
If a parallelogram is a
rhombus, then each
diagonal bisects a pair
of opposite angles.
Since a rhombus is a parallelogram, a rhombus also has all the properties of parallelograms.
ABCD is a rectangle. Find each length.
1. BD
________________________
3. AC
________________________
2. CD
_________________________
4. AE
_________________________
KLMN is a rhombus. Find each measure.
5. KL
________________________
6. m∠MNK
_________________________
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
6-30
Holt McDougal Geometry
Name ________________________________________ Date ___________________ Class __________________
LESSON
6-4
Reteach
Properties of Special Parallelograms continued
A square is a quadrilateral with four right angles and four congruent sides.
A square is a parallelogram, a rectangle, and a rhombus.
Show that the diagonals of square HJKL are congruent
perpendicular bisectors of each other.
Step 1
Show that HK ≅ JL .
HK =
JL =
(6 − 0)
( 4 − 2)
2
2
+ ( 4 − 2 ) = 2 10
2
+ ( 0 − 6 ) = 2 10
2
HK = JL = 2 10, so HK ≅ JL .
Step 2
Show that HK ⊥ JL .
4−2 1
=
slope of HK =
6−0 3
slope of JL =
0−6
= −3
4−2
Since the product of the slopes is −1, HK ⊥ JL .
Step 3
Show that HK and JL bisect each other by comparing their midpoints.
midpoint of HK = (3, 3)
midpoint of JL = (3, 3)
Since they have the same midpoint, HK and JL bisect each other.
The vertices of square ABCD are A(−1, 0), B(−4, 5), C(1, 8), and D(4, 3).
Show that each of the following is true.
7. The diagonals are congruent.
_________________________________________________________________________________________
8. The diagonals are perpendicular bisectors of each other.
_________________________________________________________________________________________
_________________________________________________________________________________________
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
6-31
Holt McDougal Geometry
6-5 CONDITIONS FOR SPECIAL
Reteach
1. 13 in.
PARALLELOGRAMS
2. 5 in.
3. 13 in.
4. 6.5 in.
5. 28
6. 50°
7. AC = BD = 2 17 , so AC ≅ BD .
1
,
4
so AC ⊥ BD . The mdpts. of AC and BD
8. Slope of AC = 4 and slope of BD = −
are at (0, 4), so AC and BD bisect each
other.
Challenge
1.
Practice A
1.
3.
5.
7.
9.
11.
12.
rhombus
2.
rectangle; rhombus 4.
rhombus
6.
sides
8.
parallelogram
10.
rhombus
rectangle; rhombus
perpendicular
diagonals
rectangle
congruent
rectangle
Practice B
1. Possible answer: To know that the
reflecting pool is a parallelogram, the
congruent sides must be opposite each
other. If this is true, then knowing that
one angle in the pool is a right angle or
that the diagonals are congruent proves
that the pool is a rectangle.
2. Not valid; possible answer: you need to
know that . AC ⊥ BD .
1. 8 yd
2. 25.3 ft
3. possible answer: you need to know that
AC and BD bisect each other.
4. valid
5. Not valid; possible answer: you need to
know that AD || BC .
3. 106°
4. 2.6 m
6. rectangle, rhombus, square
Problem Solving
5. 13 in. by 12
1
in.
4
26 ;
1
−5;
5
6. B
7. F
26
7. rhombus
Reading Strategies
1. no
2. yes
3. yes
4. no
5. no
2;3 2
1; −1
Practice C
6. They are all polygons, and they all have 4
sides.
1. Parallelogram and rhombus; possible
answer: in a square or a rectangle, the
interior angles must measure 90°.
Therefore the longest side of the triangle
formed by two sides and a diagonal must
be the diagonal.
7. All 4 angles would have to be right
angles.
8. All 4 sides would have to be congruent.
2. rhombus
3. x 3
4. 60° and 120°
5. 3
6. 1
7. 1
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
A64
Holt McDougal Geometry