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6.3 Proving Quadrilaterals are Parallelograms

California State Standards for Geometry4: Prove basic theorems involving congruence & similarity.7: Prove & use theorems involving parallel lines and properties of quadrilaterals.12: Find & use side and angle measures in triangles and polygons17: Prove theorems using coordinate geometry

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If a quadrilateral has both pairs of opposite sides

are congruent,

AB DCAD BC

||AB DC

||AD BC

theorem

A B

CD

parallelogra

m

then it is a parallelogram

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theorem

If both pairs of opposite angles of a quadrilateral

are congruent,

A B

CD

A C B D

||AB DC

||AD BCparallelogra

m

then it is a parallelogram

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180m A m B 180m A m D

||AB DC

||AD BC

theorem

If a quadrilateral has one angle that issupplementary to both of its consecutive

angles,

A B

CD

parallelogra

m

then it is a parallelogram.

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AM CM||AB DC

theorem

if a quadrilateral’s diagonals bisect each other,

A B

CD

M

BM DM||AD BC

parallelogra

m

then it is a parallelogram

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AD BC

||AB DC||AD BC

A B

CD

theorem

If a quadrilateral has one pair of opposite sides

parallel and congruent,

AB DCparallelogra

m

then it is a parallelogram

||AD BC

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Example

Decide whether you are given enough information to determine that thequadrilateral is a parallelogram. Explain your reasoning.

If a quadrilateral has one angle that is supplementary to both of its consecutive

angles,then it is a parallelogram.

YES

created by chris markstrum 2005

Example

Decide whether you are given enough information to determine that thequadrilateral is a parallelogram. Explain your reasoning.

If both pairs of opposite angles

of a quadrilateral are congruent,

then it is a parallelogram.

YES

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Example

Decide whether you are given enough information to determine that thequadrilateral is a parallelogram. Explain your reasoning.

Definition of a parallelogram

YES

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Example

Describe how you would prove that ABCD is a parallelogram.

If a quadrilateral has one pair of opposite sides

parallel and congruent, then it is a parallelogram.

Congruent alternate interior angles

make parallel lines.||BC AD

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Example

Definition of parallelogram

Congruent alternate interior angles

make parallel lines.||BC AD

Describe how you would prove that ABCD is a parallelogram.

||AB CD

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Example

Describe how you would prove that ABCD is a parallelogram.

Definition of parallelogram

Congruent alternate interior angles

make parallel lines.

||BC AD

||AB CD

Congruent corresponding angles

make parallel lines.

created by chris markstrum 2005Given:

Prove: PQRS is a parallelogram P Q S is supplementary to and

3. is supplementary to P S

1. is supplementary to P Q

Statement Reason

2. ||PS QR

4. ||PQ SR

5. is a parallelogramPQRS

2. Supplementary consecutive interior

angles means the lines are parallel.

1. Given

3. Given

4. Supplementary consecutive interior

angles means the lines are parallel.

5. Definition of a parallelogram

created by chris markstrum 2005Given:

Prove: ABCD is a parallelogram

, ||AB CD AB CD A B

CD

3. CAB ACD

1. Given1. , ||AB CD AB CD

Statement Reason

2. Reflexive2. AC AC

3. || . . 'a i s

4. SAS 4. CAB ACD 5. ACB CAD 5. C.P.C.T.C.

7. is a parallelogramABCD

6. . . ' ||a i s 6. ||AD BC

7. Def parallelogram

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is a parallelogram

PQT RSTPQRS

Given:Prove:

P Q

RS

T

3. ||PQ SR

1. Given1. PQT RST Statement Reason

2. C.P.C.T.C.2. ,PQ RS PQT RST

3. . . ' ||a i s

4. opp. sides and ||4. is a parallelogramPQRS

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To get a parallelogram Show opposite sides || Show opposite sides Show opposite angles Show consecutive

angles are supplementary Show diagonals bisect each other Show one pair of

opposite sides are both || and

( – 1, 2) A

C (1, – 2)

x

y

( 3, 2)B

D( – 3, – 2)

Show that quadrilateral ABCD is a parallelogram

Which of

the

theorems

can be

used?

1

1

–3

–4

Not on the coordinate

plane.

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x

y

202 22 4 223 1 2 2

2 22 2 1 3 2 24 22 520

( – 1, 2) A

C (1, – 2)

( 3, 2)B

D( – 3, – 2)

Show that quadrilateral ABCD is a parallelogram

1

1

–3

–4

ABm Show opposite sides ||

DCm

ADm

BCm

00

22

||AB DC ||AD BC

AB 4

4

2 5

AB DC

AD BC

Show opposite sides

DC

AD

BC

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Summary

Six ways to show a quadrilateral is a parallelogram1) Show two pairs of opposite parallel sides.

2) Show two pairs of opposite congruent sides.

3) Show two pairs of opposite congruent angles.

4) Show one angle is supplementary to both consecutive angles.

5) Show the diagonals bisect each other.

6) Show that one pair of opposite sides are both congruent and parallel.