Suppose the line shown is translated 2 units to the left and 1 unit down. Which point would lie on the translated line ?

a. (-2,-2)

b. (-1,1)

c. (0,2)

d. (2,3)

5.1 warm-up

6

-3 -2 -1 0 1 2 3 x

-3

12

3 y

6.3 Proving that a Quadrilateral is a Parallelogram

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What makes a quadrilateral a parallelogram?

• 1. Opposite sides are congruent

• 2. Opposite angles are congruent

• 3. Diagonals bisect each other

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Now, we look at some theorems

• If both pairs of opposite sides of If both pairs of opposite sides of a quadrilateral are congruent, a quadrilateral are congruent,

then the quadrilateral is a then the quadrilateral is a parallelogram.parallelogram.

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Now, we look at some theorems

• If both pairs of opposite angles If both pairs of opposite angles of a quadrilateral are congruent, of a quadrilateral are congruent,

then the quadrilateral is a then the quadrilateral is a parallelogram.parallelogram.

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Now, we look at some theorems

• If the diagonals of a quadrilateral If the diagonals of a quadrilateral bisect each other, then the bisect each other, then the

quadrilateral is a parallelogram,quadrilateral is a parallelogram,

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Now, we look at some theorems

• If the one pair of opposite sides If the one pair of opposite sides of a quadrilateral are both of a quadrilateral are both

congruent and parallel, then the congruent and parallel, then the quadrilateral is a parallelogram.quadrilateral is a parallelogram.

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Lets try a proofLets try a proof

• Given: ABDCDB, BDADBC, AC• Prove: ABCD is a parallelogram

A B

CD

Statement Reason

ABCD is a parallelogram

If opposite ’s ,

then parallaogram

•Given

ABD+CBDCDB+ADB

Addition

ABDCDB,

BDADBC

AC

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OK, here comes a problem.OK, here comes a problem.

A

B

C

D

3X

X+3.2

Y+2

3Y

ABCD is a parallelogram. Solve for X & Y.

3X = X+3.2 3Y = Y+2- 1X - 1X

2X = 3.2

2 2

X = 1.6

- 1Y - 1Y

2Y = 2

2 2Y = 1

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Just one more problemJust one more problem

A

B

C

D

m+9.1

4k-3.5

2.6k

m

ABCD is a parallelogram. Solve for m & k.

m+9.1 = 4k-3.5 m = 2.6k

substitution

2.6k+9.1 = 4k-3.5- 4k - 4k

-1.4k+9.1 = -3.5

- 9.1 - 9.1

-1.4k = -12.6

-1.4 -1.4

k = 9

m = 2.6(9)

m = 23.4

substitution

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And now the assignment.And now the assignment.

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AssignmentWorkbook

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