Download Chapter 6 Lesson 4

Chapter 6 Lesson 4
Objective: To use properties
of diagonals of rhombuses and
rectangles.
Rhombuses
Theorem 6-9
Each diagonal of a rhombus bisects two
angles of the rhombus.
1  2
3  4
1  2  3  4
Theorem 6-10
The diagonals of a rhombus are
perpendicular.
AC  BD
Example 1: Finding Angle Measures
MNPQ is a rhombus and mN = 120.
Find the measures of the numbered
angles.
m1  m3
Isosceles ∆ Theorem
m1  m3  120  180 ∆ Angle-Sum Theorem
2(m1)  120  180
2(m1)  60 m1  m2  m3  m4
m1  30
Example 2:
Finding Angle Measures
Find the measures of the
numbered angles in the
rhombus.
m1  90
m3  50
m2  50
Theorem 6-10
Theorem 6-9
Theorem 6-9
m2  m4  90  180
50  m4  90  180
140  m4  180
m4  40
Rectangles
Theorem 6-11
The diagonals of a rectangle are congruent.
AC  BD
Example 3: Finding the Lengths of Diagonals
Find the length of the diagonals of
rectangle GFED if FD = 2y + 4 and
GE = 6y − 5.
FD  GE
Theorem 6-11
2y  4  6y  5
9  4y
9
17

9
FD  GE  2   4 
y
4
2

4
Example 4:
Finding the Lengths of Diagonals
Find the length of the diagonals of GFED
if FD = 5y – 9 and GE = y + 5.
FD  GE
Theorem 6-11
5y  9  y  5
14  4y
14
y
4
7
FD  GE     5  8.5
2
Is the
parallelogram a
rhombus or a
rectangle?
Theorem 6-12
If one diagonal of a parallelogram bisects two
angles of the parallelogram, then the
parallelogram is a rhombus.
Theorem 6-13
If the diagonals of a parallelogram are
perpendicular, then the parallelogram is a
rhombus.
Theorem 6-14
If the diagonals of a parallelogram are congruent,
then the parallelogram is a rectangle.
Example 5:
Recognizing Special Parallelograms
Determine whether the quadrilateral can be a
parallelogram. If not, write impossible.
Example 6:
Recognizing Special Parallelograms
A diagonal of a parallelogram bisects two
angles of the parallelogram. Is it possible for
the parallelogram to have sides of length 5, 6,
5, and 6?
No; if one diagonal bisects two angles, then
the figure is a rhombus and cannot have two
non-congruent sides.
Assignment
Pg.315
#1-21; 45-50;
57-60