Parallelograms and Rectangles

Quadrilateral Definitions

Parallelogram: opposite sides are parallel

Rectangle: adjacent sides are

perpendicular

the first proof…

Prove: If it is a parallelogram, then the opposite sides are

equal.By definition, a parallelogram has opposite sides that are parallel.

Construct a segment:

D

B C

A

AC

We may use the properties of parallel lines to show certain angle congruencies.

As they are alternate interior angles,

and using the reflexive property, we know

D

B C

A

DCABACandDACBCA

CACA

Therefore, we know that the following triangles are congruent because of

ASA

Since these are congruent triangles, we may assume that

Therefore, if it is a parallelogram, then the opposite sides are equal.

D

B C

A

DCABAC

DCBAandDABC

Prove: If the opposite sides are equal, then it is a

parallelogram.

Given:

Construct segment

D

B C

A

DCBAandDABC

AC

Using the reflexive property, we can say

Therefore, using SSS we know

D

B C

A

ACAC

CDAABC

As the triangles are congruent, we know that corresponding angles are congruent.

Therefore,

If the alternate interior angles are congruent, then segments

D

B C

A

DACBCAandDCABAC

DCBAandDABC

Therefore, if the opposite segments are equal, then it is

a parallelogram.

D

B C

A

Therefore…

It is a parallelogram, if and only if the opposite sides are equal.

the second proof...

Prove: If it is a parallelogram, then the

diagonals bisect each other.Given parallelogram ABCD,

Using the property proven in the previous proof,

D

B C

A

BCADandDCAB

Construct segment BD

This forms two congruent triangles,

because of SSS, as the following segments are congruent:

This implies corresponding angles are congruent

D

B C

A

CDBABD

BDBDADBCCDAB ,,

Construct segment AC

This also forms two congruent triangles

Because of SSS, as the following sides are congruent

This implies that corresponding angles are congruent

D

B C

A

CDAABC

ACACADBCCDAB ,,

Look at both diagonals and the created triangles

With both diagonals displayed, we may conclude that we have two sets of congruent triangles, based upon ASA.For example,

since

E

D

B C

A

DAEBCE

EADECBDABCEDAEBC ,,

Since we have congruent triangles

We can then say that

Therefore, the diagonals of the parallelogram bisect each other since

the segments are congruent.

DAEBCE

E

D

B C

A

AECEandDEBE

Prove:If the diagonals bisect each other, then it is a

parallelogram.

Since the diagonals bisect each other, we know certain segments are congruent.

We may also say that vertical angles are congruent

E

D

B C

A

AECEandDEBE

DECBEAandCEBAED

Using SAS, we may say there are two sets of congruent triangles

Therefore, we may say

Therefore, since the diagonals bisect each other, then the opposite sides are congruent. From the previous proof, we know that it is a parallelogram

E

D

B C

A

DECBEAandDAEBCE

BCADandDCAB

therefore,

It is a parallelogram, if and only if the diagonals bisect each other.

the third proof…

Prove: If it is a rectangle, then it is a parallelogram and the diagonals are

equal.

By definition, a rectangle has adjacent sides that are perpendicular.

Since segment BC and segment AD are both perpendicular to segment AB, we may conclude that segment BC and segment AD are parallel.

The same may be concluded about segments AB and DC.

DA

B C

Since opposite sides are parallel, we may conclude that the rectangle is also a parallelogram.

Since it is a parallelogram, then we know that opposite sides are congruent.

DA

B C

Construct Segments AC and BD

Since the rectangle is also a parallelogram, then we may say,

With the constructed segments, the congruent sides, and the right angles, we have 4 congruent triangles (by SAS):

E

DA

B C

CDABandBDAC

CDBCDAABDABC

With 4 congruent triangles, we know corresponding sides are congruent.

Therefore, we may state that:

E

DA

B C

BDAC

Hence, if it is a rectangle,

then it is a parallelogram and the diagonals are equal.

Prove: If it is a parallelogram and the diagonals are equal,

then it is a rectangle.

Given: Opposite sides of a parallelogram are both parallel and congruent.Given: The diagonals are equal.

Using SSS, we know the 4 following triangles are congruent:

E

DA

B C

CDBCDAABDABC

If the four triangles are congruent, then corresponding angles are congruent.

The sum of the angles in the parallelogram (or any quadrilateral for that matter) must be 360 degrees, and all of the interior angles must be congruent.

904

360

If the interior angles are 90 degrees, then we can say that the adjacent sides are perpendicular.

Therefore, it is a rectangle.

THEREFORE…

It is a rectangle, if and only if it is a parallelogram and the diagonals are

equal.

Parallelograms, Trapezoids, Rectangles, Rhombi, Kites, and

Squares….Oh MY!

Rectangle

Paralleogram

TrapezoidSquare Kite

Quadrilateral

Rhombus