Learn – Geometry – Quadrilaterals

1 Parallelograms
- 1.1 Exercises
  - 1.1.1 Problem set
  - 1.1.2 Problem set
2 Mid-point theorem applications
- 2.1 Exercises
  - 2.1.1 Problem set
  - 2.1.2 Problem set

Parallelograms

Exercises

Problem set

Prove each of the following theorems.

The diagonals of a parallelogram divide the parallelogram into two congruent triangles.
The opposite sides of a parallelogram are congruent.
A quadrilateral in which opposite sides are congruent is a parallelogram.
A quadrilateral in which one pair of opposite sides is parallel and congruent is a parallelogram.
The opposite angles of a parallelogram are congruent.
A quadrilateral in which opposite angles are congruent is a parallelogram.
The diagonals of a parallelogram bisect each other.
A quadrilateral in which the diagonals bisect each other is a parallelogram.
The diagonals of a rhombus bisect the angles.
A parallelogram in which the diagonals bisect the angles is a rhombus.
The diagonals of a rhombus are perpendicular to each other.
A parallelogram in which the diagonals are perpendicular is a rhombus.
The diagonals of a rectangle are congruent.
A parallelogram in which the diagonals are congruent is a rectangle.

Problem set

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

Mid-point theorem applications

Exercises

Problem set

$ABCD$ 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.
$ABCD$ 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.
$ABCD$ 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.
$ABCD$ 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.

Problem set

In , the mid-points of sides and are and respectively. If area of is square units, calculate the area of:
1. $\Delta ADF$
2. $\Delta EFC$
3. Quadrilateral $BDFE$