Lesson Notes 6.4 Special Parallelograms
Diagonals have special properties in the rectangle and rhombus.
Recall: Rhombus:
Rectangle:
If a parallelogram is:
A rhombus
Each diagonal bisects two
angles
m1  m2 and m3  m4
A rhombus
The diagonals are
perpendicular
BD  AC
A rectangle
The diagonals are
congruent
AC = BD
Guided Problem Solving
Given the rectangle to the right,
Find the missing angles.
Property
Conclusion
4 is supplementary to ___________
m4 = 180 - ______ =______
The diagonals of a rectangle are ______________
_________ = _________
The diagonals of a parallelogram ________________ each other.
BM = ___________
And AM=______
The bisected segments are congruent
BM = _____
BMC is an isosceles triangle, so by the Isosceles Triangle Theorem.
(use this theorem for any rectangle)
Triangle Angle Sum, m1  m2  40  180
3 and 2 are complementary
m1  m _________
m1  m2  ______
m3  90  _____
Note: Parallelogram -> bisects ___________ and Rectangle -> has congruent ___________
Therefore, the segments of the diagonal are _____________________ to each other.
For each parallelogram, (a) choose the best name, and then (b) find the measures of the numbered angles.
1.
2.
3.
4.
5.
6.
HIJK is a rectangle. Find the value of x and the length of each diagonal.
10.
and
11.
and
For each rhombus, (a) find the measures of the numbered angles, and then (b) find the area.
14.
15.
16.
To determine if a parallelogram is a rectangle or rhombus.
Given a parallelogram….
If this is true
Theorem 6-12
One diagonal __________
two angles of the
parallelogram
Theorem 6-6
Diagonals of a
parallelogram are
__________________
Theorem 6-7
Diagonals of a
parallelogram are
__________________
diagram
algebra
<XWE=<ZWE and
<XEY=<ZEY
OR
Then WXYZ is a
rhombus
XZ is perpendicular to YW
Then WXYZ is a
rhombus
XZ=WY
Then WXYZ is a
rectangle
Determine whether the quadrilateral can be a parallelogram. If not, write impossible.
17. One pair of opposite sides is parallel, and the other pair is congruent.
Explain your answer.
18. Opposite angles are congruent and supplementary, but the quadrilateral is not a rectangle.
The parallelograms below are not drawn to scale. Can the parallelogram have the conditions marked? If not, write
impossible. Explain your answer.
7.
8.
9.