# Parallelogram and trapezoid

*Parallelogram. Properties of a parallelogram.*

Signs of a parallelogram. Rectangle. Rhombus.

Square. Trapezoid. Isosceles trapezoid.

Midline of a trapezoid and a triangle.

Signs of a parallelogram. Rectangle. Rhombus.

Square. Trapezoid. Isosceles trapezoid.

Midline of a trapezoid and a triangle.

**
Parallelogram
**
( ABCD, Fig.32 ) is a quadrangle, opposite sides of which are two-by-two parallel.

Any two opposite sides of a parallelogram are called
*
bases
*
, a distance between them is called a
*
height
*
( BE, Fig.32 ).

**
Properties of a parallelogram.
**

1.
*
Opposite sides of a parallelogram are equal
*
( AB = CD, AD = BC ).

2.
*
Opposite angles of a parallelogram are equal
*
(
A =
C,
B =
D ).

3.
*
Diagonals of a parallelogram are divided in their intersection point into two
*

( AO = OC, BO = OD ).

4.
*
A sum of squares of diagonals is equal to a sum of squares of four sides
*
:

AC
²
+ BD
²
= AB
²
+ BC
²
+ CD
²
+ AD
²
.

**
Signs of a parallelogram.
**

A quadrangle is a parallelogram, if one of the following conditions takes place:

1.
*
Opposite sides are equal two-by-two
*
( AB = CD, AD = BC ).

2.
*
Opposite angles are equal two-by-two
*
(
A =
C,
B =
D ).

3.
*
Two opposite sides are equal and parallel
*
( AB = CD, AB || CD ).

4.
*
Diagonals are divided in their intersection point into two
*
( AO = OC, BO = OD ).

**
Rectangle.
**

If one of angles of parallelogram is right, then all angles are right (why ?). This parallelogram is called a
*
rectangle
*
( Fig.33 ).

*
Main properties of a rectangle.
*

Sides of rectangle are its heights simultaneously.

*
Diagonals of a rectangle are equal:
*
AC = BD.

*
A square of a diagonal length is equal to a sum of squares of its sides’ lengths
*
( see
above Pythagorean theorem
):

**
Rhombus.
**
If all sides of parallelogram are equal, then this parallelogram is called a

*rhombus*( Fig.34 ) .

*
Diagonals of a rhombus are mutually perpendicular
*
( AC
BD )
*
and divide
its angles into two
*
(
DCA =
BCA,
ABD =
CBD etc. ).

**
Square
**
is a parallelogram with right angles and equal sides ( Fig.35 ).

*A square is a particular case of a rectangle and a rhombus simultaneously; so, it has all their above mentioned properties.*

**
Trapezoid
**
is a quadrangle, two opposite sides of which are parallel (Fig.36).

Here AD
**
||
**
BC. Parallel sides are called
*
bases
*
of a trapezoid, the two others ( AB and CD ) –
*
lateral sides.
*
A distance between bases
(BM) is a
*
height.
*
The segment EF, joining midpoints E and F of the lateral sides, is called a
*
midline
*
of a trapezoid.

*
A midline of a trapezoid is equal to a half-sum of bases:
*

*
and parallel to them:
*
EF || AD and EF || BC.

A trapezoid with equal lateral sides ( AB = CD ) is called an
*
isosceles
*
trapezoid.
*
In an isosceles trapezoid angles by each base, are equal
*
(
A =
D,
B =
C ).
A parallelogram can be considered as a particular case of trapezoid.

**
Midline of a triangle
**
is a segment,

*joining midpoints of lateral sides*of a triangle.

*A midline of a triangle is equal to half of its base and parallel to it.*This property follows from the previous part, as triangle can be considered as a limit case (“degeneration”) of a trapezoid, when one of its bases transforms to a point.