6. Show that consecutive angles are supplementary

6. Show that consecutive angles are supplementary

What do you see?

What Makes a Quadrilateral a Parallelogram?

Are both pairs of opposite sides parallel?

Is one pair of opposite sides congruent and parallel?

In This Picture…

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.

Two pairs of opposite sides are equal.

Therefore, the quadrilateral is a parallelogram. By theorem 8.7

ANSWER

GUIDED PRACTICE for Examples 2 and 3

What theorem can you use to show that the quadrilateral is a parallelogram?4.

By theorem 8.8, if the opposite angles are Congruent, the quadrilateral is a parallelogram.

ANSWER

GUIDED PRACTICE for Examples 2 and 3

For what value of x is quadrilateral MNPQ a parallelogram? Explain your reasoning.

5.

SOLUTION

[ Diagonals in bisect each other ] By Theorem 8.62x = 10 – 3x

Add 3x to each side5x = 10Divide each side by 5x = 2

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

SOLUTIONOne 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

Show that quadrilateral ABCD is a parallelogram.

EXAMPLE 4 Use coordinate geometry

Because AB = CD = 29 , AB CD.Then use the slope formula to show that AB CD.

Slope of AB =5 – (3)

2 – (–3) = 25 Slope of CD = 2 – 0

5 – 0 = 25

Because AB and CD have the same slope, they are parallel.

AB and CD are congruent and parallel. So, ABCD is a parallelogram by Theorem 8.9.

ANSWER

EXAMPLE 4GUIDED 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 4GUIDED PRACTICE for Example 4

=[-4 – (0)]2 + (1 – 8)2

=[4 – (8)]2 + (-1 – 6)2

65

65

DK =

TA =

D

A

T

K DK and TA are congruent and parallel. So, TDKA is a parallelogram by Theorem 8.9.

In Conclusion…

• Pg 526 # 1-3, 11-14

Don’t forget your homework.