Download 6.3 Practice with Examples

LESSON
6.3
NAME _________________________________________________________ DATE ____________
Practice with Examples
For use with pages 338–346
GOAL
Prove that a quadrilateral is a parallelogram and use coordinate geometry
with parallelograms
Theorem 6.6
If both pairs of opposite sides of a quadrilateral are congruent, then the
quadrilateral is a parallelogram.
Chapter 6
Theorem 6.7
If both pairs of opposite angles of a quadrilateral are congruent, then the
quadrilateral is a parallelogram.
Theorem 6.8
If an angle of a quadrilateral is supplementary to both of its consecutive
angles, then the quadrilateral is a parallelogram.
Theorem 6.9
If the diagonals of a quadrilateral bisect each other, then the quadrilateral
is a parallelogram.
Theorem 6.10
If one pair of opposite sides of a quadrilateral are congruent and parallel,
then the quadrilateral is a parallelogram.
Ways to Prove a Shape is a Parallelogram
• Show that both pairs of opposite sides are parallel.
• Show that both pairs of opposite sides are congruent.
• Show that both pairs of opposite angles are congruent.
• Show that one angle is supplementary to both consecutive angles.
• Show that the diagonals bisect each other.
• Show that one pair of opposite sides are congruent and parallel.
106
Geometry
Practice Workbook with Examples
Copyright © McDougal Littell Inc.
All rights reserved.
LESSON
6.3
CONTINUED
EXAMPLE 1
NAME _________________________________________________________ DATE ____________
Practice with Examples
For use with pages 338–346
Using Properties of Parallelograms
Show that A2, 0, B3, 4, C2, 6, and
D3, 2 are the vertices of a parallelogram.
y
C
SOLUTION
B
There are many ways to solve this problem.
Slope of AB 40
4
32
Slope of CD 26
4
4
3 2 1
Slope of BC 64
2
2
2 3 5
5
Slope of DA 02
2
2
2 3
5
5
2
Chapter 6
Method 1 Show that opposite sides have
the same slope, so they are parallel.
D
A
1
x
AB and CD have the same slope, so they are parallel. Similarly, BC DA.
Because opposite sides are parallel, ABCD is a parallelogram.
Method 2
Show that the opposite sides have the same length.
AB 3 22 4 02 17
CD 3 22 2 62 17
BC 2 3)2 6 42 29
DA 2 32 0 22 29
AB CD and BC DA. Because both pairs of opposite sides are
congruent, ABCD is a parallelogram.
Method 3 Show that one pair of opposite sides is congruent and
parallel. Find the slopes and lengths of AB and CD as shown in Methods
1 and 2.
Slope of AB Slope of CD 4
AB CD 17
AB and CD are congruent and parallel, so ABCD is a parallelogram.
Copyright © McDougal Littell Inc.
All rights reserved.
Geometry
Practice Workbook with Examples
107
LESSON
6.3
CONTINUED
NAME _________________________________________________________ DATE ____________
Practice with Examples
For use with pages 338–346
Exercises for Example 1
Refer to the methods demonstrated in Example 1 to show
that the quadrilateral with the given vertices is a
parallelogram.
1. Show that the quadrilateral with vertices A3, 0, B2, 4,
Chapter 6
C7, 6, and D8, 2 is a parallelogram using Method 1
from Example 1.
2. Show that the quadrilateral with vertices A4, 1, B1, 2,
C4, 4, and D1, 5 is a parallelogram using Method 2
from Example 1.
3. Show that the quadrilateral with vertices A0, 6, B4, 5,
C6, 3, and D2, 2 is a parallelogram using Method 3 from
Example 1.
4. Show that the quadrilateral with vertices A1, 2, B5, 3,
C6, 6, and D0, 7 is a parallelogram using any of the three
methods demonstrated in Example 1.
108
Geometry
Practice Workbook with Examples
Copyright © McDougal Littell Inc.
All rights reserved.