Show that the diagonals of a rhombus divide it into four congruent triangles.

1. Quadrilaterals:

(i) The sum of the angles of a quadrilateral is ${360}^{°}$.

(ii) A quadrilateral is a parallelogram if its opposite sides are equal.

(iii) A quadrilateral is a parallelogram if its opposite angles are equal.

(iv) The diagonals of a quadrilateral bisect each other if it is a parallelogram.

(v) A quadrilateral is a parallelogram if one pair of opposite sides are equal and parallel.

(vi) The quadrilateral formed by joining the mid-points of the sides of a quadrilateral, in order, is a parallelogram.

2. Parallelogram:

(i) A diagonal of a parallelogram divides it into two congruent triangles.

(ii) Two opposite angles of a parallelogram are equal.

(iii) The diagonals of a parallelogram bisect each other.

(iv) In a parallelogram, the bisectors of any two consecutive angles intersect at a right angle.

(v) If a diagonal of a parallelogram bisects one of the angles of the parallelogram it also bisects the second angle.

(vi) The angles bisectors of a parallelogram form a rectangle.

(vii) Diagonals of a parallelogram are equal if and only if it is a rectangle.

(viii) Diagonals of a parallelogram are perpendicular if and only if it is a rhombus.

(ix) If the diagonals of a parallelogram are equal and intersect at right angles, then it is a square.

3. Rectangle and Square:

(i) Each of the four angles of a rectangle is a right angle.

(ii) ​​​​​​​The diagonals of a rectangle are of equal length.

(iii) ​​​​​​​The diagonals of a square are equal and perpendicular to each other.

4. Rhombus:

(i) Each of the four sides of a rhombus is of the same length.

(ii) The diagonals of a rhombus are perpendicular to each other.

5. The Mid-point Theorem:

(i) The line segment joining the mid-points of any two sides of a triangle is parallel to the third side and equal to half of it.

(ii) A line through the mid-point of a side of a triangle parallel to another side bisects the third side.