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Do not worry about your difficulties in mathematics, I assure you that mine are greater
". Einstein, Albert (1879-1955)
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Parallelogram and trapezoid
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.
Opposite sides of a parallelogram are equal ( AB = CD, AD = BC ).
Opposite angles of a parallelogram are equal (
A =
C,
B =
D ).
Diagonals of a parallelogram are divided in their intersection point into two ( AO = OC, BO = OD ).
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:
Opposite sides are equal two-by-two ( AB = CD, AD = BC ).
Opposite angles are equal two-by-two (
A =
C,
B =
D ).
Two opposite sides are equal and parallel ( AB = CD, AB || CD ).
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 sidesâ€™ lengths ( see above Pythagorean theorem ):
AC
^{2}
= AD
^{2}
+ DC
^{2}
.
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 ) â€“ 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 isoscelestrapezoid. 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 (â€œdegenerationâ€) of a trapezoid, when one of its bases transforms to a point.
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