Download Geometry Lesson 6-4 Special Parallelograms.notebook

Geometry Lesson 6­4 Special Parallelograms.notebook
November 05, 2012
Section 6­4 : Special Parallelograms
Warm­Up
PACE is a parallelogram and m<PAC = 109. Complete each of the following.
7
A
C
3.
5
1. EC =
5
4.5
4.7
2. EP =
R
3. m<CEP = P
E
4. PR =
5. RE =
6. CP =
7. m<EPA = 8. m<ECA =
If you draw two congruent isosceles triangles with base PQ, you have drawn a rhombus. R
Note that <RPQ, <SPQ, <RQP, and <SQP are all congruent
This suggests Theorem 6­9.
Q
P
S
Theorem 6­9: Each diagonal of a rhombus bisects two angles of the rhombus.
Proof: 1
Geometry Lesson 6­4 Special Parallelograms.notebook
November 05, 2012
Theorem 6­10: The diagonals of a rhombus are perpendicular.
B
C
AC
BD
A
D
Theorem 6­11: The diagonals of a rectangle are congruent.
B
C
A
D
AC ≅ BD
Example 1) MNPQ is a rhombus and m<N = 120. Find the measures of the numbered angles.
N
P
3
120 o
4
1
M
2
Q
2
Geometry Lesson 6­4 Special Parallelograms.notebook
November 05, 2012
You try) Find the measures of the numbered angles in the rhombus.
50o
1
4
2
3
Example 2) Find the length of the diagonals of rectangle GFED if FD = 2y + 4 and GE = 6y ­ 5.
F
E
G
D
3
Geometry Lesson 6­4 Special Parallelograms.notebook
You try) Find the lengths of the diagonals of GFED if FD = 5y ­ 9 and GE = y + 5
F
G
November 05, 2012
E
D
Theorem 6­12: If one diagonal of a parallelogram bisects two angles of the parallelogram, then the parallelogram is a rhombus. Theorem 6­13: If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.
Theorem 6­14: If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.
4
Geometry Lesson 6­4 Special Parallelograms.notebook
November 05, 2012
Example 3)
Determine whether the quadrilateral can be a parallelogram. If not, write impossible.
a. The quadrilateral has congruent diagonals and one angle of 60 o.
b. The quadrilateral has perpendicular diagonals and four right angles.
You try) A diagonal of a parallelogram bisects two angles of the parallelogram. It is possible for the parallelogram to have sides of lengths 5, 6, 5, and 6? Explain.
5
Geometry Lesson 6­4 Special Parallelograms.notebook
November 05, 2012
Example 4) Builders use properties of diagonals to "square up" rectangular shapes like building frames and playing­field boundaries. Suppose you are on the volunteer building team. You are helping to lay out a rectangular patio. Explain how to use 2 ropes and the properties of diagonals to locate the four corners.
6