A math lesson on the parallelogram in the fifth grade. is to be followed in its entirety to progress throughout the year.
This lesson involves the following concepts:
- definition;
- property of the opposite sides;
- property of opposite angles;
- property of the diagonals;
- the rectangle, the rhombus, the rectangle and the square.
The student should be able to draw with geometry equipment (ruler, square, compass and protractor) but also know how to use the different properties of this figure concerning its opposite sides, its opposite angles and its diagonals. We will end this chapter with the study of particular parallelograms such as the rectangle, the rhombus and the square in the fifth grade.
I. The parallelogram
1.Definition
It is a quadrilateral having its opposite sides parallel two by two.
Example:
The lines (AB) and (DC) are parallel.
The lines (AD) and (BC) are parallel.
The quadrilateral ABCD is a parallelogram.
The center of symmetry is the point O which corresponds to the point of intersection of its diagonals.
Remark:
Point O is the center of symmetry of parallelogram ABCD.
The image of the segment [AB] by the symmetry of center O is the segment [DC].
The image of the angle 
by the symmetry of center O is the angle .
The diagonals of this figure intersect in the middle.
Proof:
If a quadrilateral is a parallelogram then its center of symmetry is the point of intersection of the diagonals.
By definition of the center of symmetry, we deduce that O is the middle of [AC] and O is the middle of [BD].
Therefore, the diagonals [AC] and [BD] intersect in their middle O which is the center of symmetry of the parallelogram.
Opposite sides of a parallelogram have the same length.
Proof:
If a quadrilateral is a parallelogram then its center of symmetry is the point of intersection of the diagonals.
The symmetric of the segment [AB] is [DC] and the symmetric of the segment [AD] is [BC].
Central symmetry preserves the lengths of the segments so AB=DC and AD=BC.
The opposite angles of a parallelogram have the same measure.
Proof:
If a quadrilateral is a parallelogram then its center of symmetry is the point of intersection of the diagonals.
The mage of the angle by the symmetry of center O is the angle .
The image of the angle by the symmetry of center O is the angle .
Central symmetry preserves the angle measures so and .
II. particular parallelograms
A rectangle, a rhombus and a square are particular parallelograms.
A square is both a rhombus and a rectangle, it has all the properties of a rhombus and a rectangle.
Application:
Are these statements true or false?
- A parallelogram has two axes of symmetry.
- If E and F are the respective symmetries of G and H with respect to ,then EFGH is a parallelogram of center O.
- A parallelogram has four equal angles.
- If a quadrilateral has three right angles, then it is a rectangle.
- If a quadrilateral has three equal sides, then it is a rhombus.
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