Statements Reasons

Given: WXYZ is a parallelogram

Prove: ΔYZX ΔWXZW X

YZ1) Complete the Proof.

2. WX ZY, WZ YX

1. WXYX is a

3. ZX ZX

4. ΔYZX ΔWXZ

1. Given

3. Reflexive Property

2. Opposite sides of a parallelogram are congruent.

4. SSS Postulate

Statements Reasons

Given: BCDE is a parallelogram

AE CD

Prove: EAB EBA

B C

DE2) Complete the Proof.

2. EB DC

1. BCDE is a

3. AE CD

4. EB AE

1. Given

3. Given

2. Opposite sides of a parallelogram are congruent.

4. Substitution

A

5. EAB EBA5. If two sides of a triangle are , then the angles opposite those sides are .

Pg.1689) Def. of Parallelogram10) If lines are ||, alternate interior angles are congruent.11) Opposite angles of a parallelogram are congruent.12) Opposite sides of a parallelogram are congruent.13) Diagonals of a parallelogram bisect each other.14) Diagonals of a parallelogram bisect each other.

Pg.1695)a = 8, b = 10, x = 118, y = 626)a = 8, b = 15, x = 80, y = 707)a = 5, b = 3, x = 120, y = 228)a = 9, b = 11, x = 33, y = 279)a = 8, b = 8, x = 56, y = 6810) a = 10, b = 4, x = 90, y = 45

Pg.16911) Perimeter = 6012) ST = 14, SP = 13

Pg.17019) x = 3, y = 520) x = 7, y = 1821) x = 13, y = 5

1) Name all the properties of a parallelogram.

•

•

•

•

•

2 pairs of opposite sides are parallel2 pairs of opposite sides are congruent2 pairs of opposite angles are congruent

Consecutive angles are supplementary

Diagonals bisect each other

A B

CD

M

DAC _____

AB = _____

mBCD = _____

CM = _____

AD || _____

DA = _____

BAC _____

BM = _____

Given the below

parallelogram, complete the statements.

BCA

mDAB DCA

CD

AM

BC

BC

DM

Theorem 5-4: If both pairs of opposite sides of a quadrilateral are congruent,

then the quadrilateral is a parallelogram.E F

GH

PROOF OF THEOREM 5-4: E F

GH

12

34

Given: EF GH, FG EH

Prove: EFGH is a

6. Def. of parallelogram

3. ΔEFH ΔGHF 3. SSS Postulate

4. CPCTC4. 1 4, 2 3

2. Reflexive Property

5. If alternate interior angles are congruent, then lines are parallel.5. EF || GH, FG || HE

2. FH FH

1. EF GH; FG EH 1. Given

6. EFGH is a

Theorem 5-5: If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram.

E F

GH

PROOF OF THEOREM 5-5:

Given: EF GH, EF || GH

Prove: EFGH is a

6. If both pairs of opposite sides of a quadrilateral are congruent, then it is a parallelogram.

4. ΔEFH ΔGHF 4. SAS Postulate

5. CPCTC

2. 1 4

3. Reflexive Property

2. If lines are parallel, alternate interior angles are congruent.

5. FG HE

3. FH FH

1. EF GH; EF || GH 1. Given

6. EFGH is a

E F

GH

12

34

Theorem 5-6: If both pairs of opposite angles of a quadrilateral are

congruent, then the quadrilateral is a parallelogram.

D C

BA

PROOF OF THEOREM 5-6: D C

BA

x y

y x

Given: mA = mC = y;mB = mD = x

Prove: ABCD is a

6. Def. of parallelogram

3. x + y = 180 3. Division Property

4. Definition of supp. angles4. A and D are supp.

2. The sum of the interior angles of a quadrilateral is 360.

5. If same-side interior angles are supplementary, then lines are parallel.

5. AB || CD, AD || BC

2. 2x + 2y = 360

1. mA = mC = y;mB = mD = x

1. Given

6. ABCD is a

A and B are supp.

Theorem 5-7: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Q R

ST

M

Based on the markings on each figure,

A. Decide if each figure is a parallelogram (YES or NO).

B. If yes, justify your answer. State the theorem that is supported by the figure.

If no, identify which theorem is not justified or is not met by the

diagram.

EXAMPLE 1:

EXAMPLE 2:

NO

YES

Opposite sides are not

congruent.

Both pairs of opposite sides are

parallel.

EXAMPLE 3:

EXAMPLE 4:

YES

YESDiagonals bisect

each other.

Both pairs of opposite angles are congruent.

EXAMPLE 5:

EXAMPLE 6:

NO

YES

Pair of congruent/parallel

sides is not the same pair of

sides.

One pair of sides is both congruent

and parallel.