Transcript Are both pairs of opposite sides congruent?

6. Show that consecutive angles are supplementary
What Makes a Quadrilateral a Parallelogram?
What do you see?
Are both pairs of opposite
sides parallel?
In This Picture…
Is one pair of opposite sides
congruent and parallel?
Are both pairs of opposite
sides congruent?
Are both pairs of opposite
angles congruent?
What is this Picture?
What is this?
What is this?
Is a Pentagon a
Parallelogram?
NO!
GUIDED PRACTICE
for Examples 2 and 3
What theorem can you use to show that the
quadrilateral is a parallelogram?
3.
ANSWER
Two pairs of opposite sides are equal.
Therefore, the quadrilateral is a parallelogram. By
theorem 8.7
GUIDED PRACTICE
for Examples 2 and 3
What theorem can you use to show that the
quadrilateral is a parallelogram?
4.
ANSWER
By theorem 8.8, if the opposite angles are Congruent,
the quadrilateral is a parallelogram.
GUIDED PRACTICE
for Examples 2 and 3
5. For what value of x is
quadrilateral MNPQ a
parallelogram?
Explain your reasoning.
SOLUTION
2x = 10 – 3x
5x = 10
x=2
By Theorem 8.6
[ Diagonals in
bisect each other ]
Add 3x to each side
Divide each side by 5
6. Show that consecutive angles are supplementary
Game Time: Name that Theorem
Game Time: Name that Theorem
Game Time: Name that Theorem
Game Time: Name that Theorem
Game Time: Name that Theorem
6. Show that consecutive angles are supplementary
Game Time: Name that Theorem
EXAMPLE 4
Use coordinate geometry
Show that quadrilateral ABCD
is a parallelogram.
SOLUTION
One way is to show that a pair
of sides are congruent and
parallel. Then apply Theorem
8.9.
First use the Distance Formula
to show that AB and CD are
congruent.
AB =
[2 – (–3)]2 + (5 – 3)2 =
29
CD =
(5 – 0)2 + (2 – 0)2
29
=
EXAMPLE 4
Use coordinate geometry
Because AB = CD =
29 , AB
CD.
Then use the slope formula to show that AB CD.
5 – (3)
2
2 Slope of CD = 2 – 0
Slope of AB = 2 – (–3) =
=
5–0
5
5
Because AB and CD have the same slope,
they are parallel.
ANSWER
AB and CD are congruent and parallel. So, ABCD is a
parallelogram by Theorem 8.9.
EXAMPLE
4
GUIDED PRACTICE
for Example 4
6. Refer to the Concept Summary. Explain how
other methods can be used to show that
quadrilateral ABCD in Example 4 is a
parallelogram.
SOLUTION
Find the Slopes of all 4 sides and show that each
opposite sides always have the same slope and,
therefore, are parallel.
Find the lengths of all 4 sides and show that the
opposite sides are always the same length and,
therefore, are congruent.
Find the point of intersection of the diagonals and
show the diagonals bisect each other.
EXAMPLE
4
GUIDED PRACTICE
K
for Example 4
DK and TA are congruent and parallel. So,
TDKA is a parallelogram by Theorem 8.9.
A
D
T
DK =
[-4 – (0)]2 + (1 – 8)2 =
65
TA =
[4 – (8)]2 + (-1 – 6)2 =
65
In Conclusion…
Don’t forget your homework.
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