Learn – Geometry – Quadrilaterals

1 Parallelograms

1.1 Exercises

1.1.1 Problem set

Prove each of the following theorems.

  1. The diagonals of a parallelogram divide the parallelogram into two congruent triangles.
  2. The opposite sides of a parallelogram are congruent.
  3. The opposite angles of a parallelogram are congruent.
  4. A quadrilateral in which opposite sides are congruent is a parallelogram.
  5. A quadrilateral in which one pair of opposite sides is parallel and congruent is a parallelogram.
  6. A quadrilateral in which opposite angles are congruent is a parallelogram.
  7. The diagonals of a parallelogram bisect each other.
  8. A quadrilateral in which the diagonals bisect each other is a parallelogram.
  9. The diagonals of a rhombus bisect the angles.
  10. A parallelogram in which the diagonals bisect the angles is a rhombus.
  11. The diagonals of a rhombus are perpendicular to each other.
  12. A parallelogram in which the diagonals are perpendicular is a rhombus.
  13. The diagonals of a rectangle are congruent.
  14. A parallelogram in which the diagonals are congruent is a rectangle.

1.1.2 Problem set

  1. ABCD is a parallelogram. O is the point of intersection of diagonals AC and BD. If AB = BC = 4, find the measure of \angle COD.
  2. PQRS is a parallelogram. M is the point of intersection of diagonals PR and QS. If \angle PMQ = 90^\circ, PQ = 5, find QR.

2 Mid-point theorem applications

2.1 Exercises

2.1.1 Problem set

  1. ABCD is a quadrilateral. Say, E, F, G and H are mid-points of AB, BC, CD and DA respectively. Then, prove that EFGH is a parallelogram.
  2. ABCD is a rectangle. Say, E, F, G and H are mid-points of AB, BC, CD and DA respectively. Then, prove that EFGH is a rhombus.
  3. ABCD is a rhombus. Say, E, F, G and H are mid-points of AB, BC, CD and DA respectively. Then, prove that EFGH is a rectangle.
  4. ABCD is a quadrilateral. Say, E, F, G and H are mid-points of AB, BC, CD and DA respectively. Then, prove that EG and FH bisect each other.

2.1.2 Problem set

  1. In \Delta ABC, the mid-points of sides AB, BC and CA are D, E and F respectively. If area of \Delta ABC is 12 square units, calculate the area of:
    1. \Delta ADF
    2. \Delta EFC
    3. Quadrilateral BDFE