Chapter 6Proving a Quadrilateral is a Parallelogram

quadrilateral

trapezoid

isoscelestrapezoid

parallelogram

squarerectanglerhombus

Parallelogram

Rectangle

S

R

I

Square

Rhombus

Kite

Trapezoid

Isosceles Trapezoid

Symbols

Parallelogram1. Both pairs of opposite sides are parallel.2. Both pairs of opposite sides are congruent .3. Both pairs of opposite angles are congruent.4. Diagonals bisect each other (property).5. Consecutive angles are supplementary.

Ways to Prove a Quadrilateral is a Parallelogram1. Both pairs of opposite sides are parallel.2. Both pairs of opposite sides are congruent.3. Both pairs of opposite angles are congruent.4. The diagonals bisect each other.5. One pair of opposite sides is both congruent and parallel.

State the property or definition which proves that the figure is a parallelogram.

Q

U A

D

Both pairs of opposite sides are parallel.

Both pairs of opposite sides are congruent.

Both pairs of opposite angles are congruent..

The diagonals bisect each other.

One pair of opposite sides is both congruent and parallel.

State the property or definition (if there is one) that proves that the figure is a parallelogram.

Q

U

A

DBoth pairs of opposite sides are parallel.

Both pairs of opposite sides are congruent.

Both pairs of opposite angles are congruent.

The diagonals bisect each other.

One pair of opposite sides is both congruent and parallel.

State the property or definition (if there is one) that proves that the figure is a parallelogram.

Q

U A

DD

Both pairs of opposite sides are parallel.

Both pairs of opposite sides are congruent.

Both pairs of opposite angles are congruent.

The diagonals bisect each other.

One pair of opposite sides is both congruent and parallel.

State the property or definition (if there is one) that proves that the figure is a parallelogram.

Q

U A

DBoth pairs of opposite sides are parallel.

Both pairs of opposite sides are congruent.

Both pairs of opposite angles are congruent.

The diagonals bisect each other.

One pair of opposite sides is both congruent and parallel.

State the property or definition (if there is one) that proves that the figure is a parallelogram.

Q

U A

D

Cannot be proved to be a parallelogram! The most descriptive name you could give this quadrilateral is “isosceles trapezoid.”

Both pairs of opposite sides are parallel.

Both pairs of opposite sides are congruent.

Both pairs of opposite angles are congruent.

The diagonals bisect each other.

One pair of opposite sides is both congruent and parallel.

S

T V

R X

ram.parallelog a isRSTV :Prove

TVSRSV

RSTXRV :Given

Statements Reasons

TVSRSVRSTXRV

GivenVR||TS

SR||TV

Alt. int. s z|| lines

S.S. int. s z|| lines

Corr. s z|| lines

RSTV

Both prs. opp. sides quad. ||

Both prs. opp. sides quad. z Both prs. opp. s quad. z

Diags. quad. bis. each otherz

One pr. opp. sides quad. z+||

ram.parallelog a is SMPR :Prove

O Circle :Given

M

S R

P

O

Statements Reasons

GivenORMO

OPSO

Radii of a circle z

If pt. div. seg. into 2 z segs., it bis.it

SMPR

Both prs. opp. sides quad. ||

Both prs. opp. sides quad. z Both prs. opp. s quad. z

Diags. quad. bis. each otherz

One pr. opp. sides quad. z+||

Circle O

O midpt. SP,MR

MR bis. SP ;SP bis. MR

Proving a Quadrilateral is a Parallelogram1. If both pairs of opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram (reverse of

the definition).2. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram (converse

of a property).3. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram (converse of a

property).4. If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram

(converse of a property).5. If one pair of opposite sides of a quadrilateral are both parallel and congruent, then the quadrilateral is a

parallelogram.

Proving a Quadrilateral is a RectangleFirst, prove the quadrilateral is a parallelogram and then one of the following:1. If a parallelogram contains at least one right angle, then it is a rectangle (reverse of the definition).2. If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.Or, without first proving a parallelogram, prove the following:1. If all four angles of a quadrilateral are right angles, then it is a rectangle.

Proving a Quadrilateral is a KiteProve one of the following:• If two disjoint pairs of consecutive sides of a quadrilateral are congruent, then it is a kite (reverse of the

definition).• If one of the diagonals of a quadrilateral is the perpendicular bisector of the other diagonal, then the quadrilateral

is a kite.

Proving a Quadrilateral is a RhombusFirst , prove the quadrilateral is a parallelogram and then one of the following:1. If a parallelogram contains a pair of consecutive sides that are congruent, then it is a rhombus (reverse of the

definition).2. If either diagonal of a parallelogram bisects two angles of the parallelogram, then it is a rhombus.Or, without first proving a parallelogram, prove the following:1. If the diagonals of a quadrilateral are perpendicular bisectors of each other, then the quadrilateral is a rhombus.

Proving a Quadrilateral is a SquareIf a quadrilateral is both a rectangle and a rhombus, then it is a square (reverse of the definition).

Proving a Trapezoid is IsoscelesProve one of the following:1. If the non-parallel sides of a trapezoid are congruent, then it is isosceles (reverse of the definition).2. If the lower or the upper base angles of a trapezoid are congruent, then it is isosceles.3. If the diagonals of a trapezoid are congruent, then it is isosceles.

A B

CD

A B

CD

A

B

C

D

A B

CD

A B

CD

A B

CD

A

D C

B2x + 5

3x - 227

4x - 17

rhombus. a NOT is ABCD that show ,DCAB If

x

x

xx

CDAB

11

222

17452 27312332)11(323

275225)11(252

xBC

xAB

All sides NOT congruent; hence ABCD is NOT a rhombus!

be? x of value

the must what rectangle, a be to RECT for order In

R

C

E

T42

(2x+6)

21 x

42 2x

90 48 2x

90 42 6)(2x Hence,

. rt. a be must RTC

A

D C

B

Parallelogram: If one pair of opposite sides of a quadrilateral are both parallel and congruent, then it is a parallelogram.

What is the most descriptive name for this quadrilateral?

A

D C

B

Isosceles Trapezoid: By definition a trapezoid has exactly one pair of parallel sides; it is isosceles because the legs are congruent.

What is the most descriptive name for this quadrilateral?

A

D C

B

Rectangle: It is a parallelogram because the diagonals of the quadrilateral bisect each other; furthermore, it is a rectangle because the diagonals are congruent.

What is the most descriptive name for this quadrilateral?

A

D C

B

Kite: If one of the diagonals of a quadrilateral is the perpendicular bisector of the other diagonal, then the quadrilateral is a kite.

What is the most descriptive name for this quadrilateral?

A

D C

B

Trapezoid: By definition, a trapezoid has exactly one pair of parallel sides. (AD || BC: If two coplanar lines are perpendicular to a third line, DC, then they are parallel.)

What is the most descriptive name for this quadrilateral?

8 10

A

D C

B

Rectangle: It is a parallelogram because both pairs of opposite angles are congruent. Since it is a parallelogram, consecutive angles are supplementary. By the given information, consecutive angles are congruent. If two angles are both congruent and supplementary, then they are right angles. Finally, it is a rectangle because all angles are right angles.

What is the most descriptive name for this quadrilateral?

A

D C

B

Rhombus: It is a parallelogram because the diagonals of the quadrilateral bisect each other; furthermore, it is a rhombus because the diagonals are perpendicular bisectors of each other.

What is the most descriptive name for this quadrilateral?

A

D C

B

Quadrilateral: There is NOT enough information to classify it as anything more specific.

What is the most descriptive name for this quadrilateral?

A (1, 1)

D (10,3)

C (8, 10)

B (2, 6)

M, N, O, and P are midpoints of the sides of ABCD. Find the coordinates of M, N, O, and P.•Find the slopes of MN and PO.•What is true about MN and PO?

x

y

O

P

N

M

x2+3x+2