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.
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 =
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:
AC² + BD²
= AB² + BC²
+ CD² + AD² .
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 (
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 ).
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 ):
= AD² + DC².
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.