Download Proving Quadrilaterals are Parallelograms

Proving Quadrilaterals are
Parallelograms
Standards #4,7,12,17
Section 6.3
Tuesday, November 13, 2012
Goals: Today you will learn to
• Prove a quadrilateral is a
parallelogram.
• Use coordinate geometry with
parallelograms.
Theorems Proving Parallelograms
Theorem 6.6:
If both pairs of opposite sides of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram.
Given:
B
C
ABCD and ADBC,
Conclusion:
A
ABCD is a parallelogram
D
Theorems Proving Parallelograms
Theorem 6.7
If both pairs of opposite angles of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram.
Given:
A C and BD
Conclusion:
ABCD is a parallelogram
C
B
A
D
Theorems Proving Parallelograms
Theorem 6.8
If an angle of a quadrilateral is
supplementary to both of its consecutive
angles, then the quadrilateral is a
parallelogram.
Given:
B is supplementary to A
and B is supplementary to C
Conclusion:
ABCD is a parallelogram
A
B
(180-x)º
xº
xº
D
C
Theorems Proving Parallelograms

Theorem 6.9
If the diagonals of a quadrilateral bisect
each other, then the quadrilateral is a
parallelogram.
B
Given: AC bisects BD
and BD bisects AC
Conclusion:
ABCD is a parallelogram
A
D
C
Theorems Proving Parallelograms

Theorem 6.10
If one pair of opposite sides of a
quadrilateral are congruent and parallel,
then the quadrilateral is a parallelogram.
B
Given: ADBC, AD ll BC
C
Conclusion:
ABCD is a parallelogram
A
D
Six Ways to Prove Quadrilaterals are
Parallelograms
1.
2.
3.
4.
5.
6.
Summary
Show both pairs of opposite sides are parallel.
Show both pairs of opposite sides are congruent.
Show both pairs of opposite angles are congruent.
Show both diagonals bisect each other.
Show one angle is supplementary to both
consecutive angles.
Show one pair of opposite sides are both
congruent and parallel.
Example #1
Given: PQT RST
Prove: PQRS is a parallelogram
Q
P
T
S
R
Example #2
Given: ABCD is a parallelogram
FE ll DC
Prove: ABEF is a parallelogram
D
C
F
A
E
B
Example #3
Given: HJKM is a parallelogram
IJKLMH
Prove: HIKL is a parallelogram
H
M
I
1
L
J
2
K
Example #4
Given:12, IJKLMH
Prove: HIKL is a parallelogram
H
M
I
1
L
J
2
K
Example #5
Show that A(-1,2), B(3,2), C(1,-2)
and D(-3,-2) are the vertices of a
parallelogram by
a. Showing both pair of opposite sides
are parallel.
b. Showing both pair of opposite sides
are congruent.
Example #6
Identify any quadrilateral that is a
parallelogram.
a. G(-3,1), H(4,1), I(3,6), J(-1,6)
b. P(-2,2), Q(1,1), R(4,4), S(1,4)
c. W(3,-1), X(4,2), Y(1,5), Z(0,2)
Homework
Page 342
# 2-7, 9-19, 45-47