In this video we will learn about the different properties of quadrilaterals.

We have learnt that any four sided closed figure is called a Quadrilateral. Each quadrilateral has two parameters, angles and sides.

Based on the sides and angles there are certain properties of quadrilaterals.  We are discussing them below:

Properties based on Length of Sides

• The Mid-Point Theorem

Joining mid-points of all sides of a quadrilateral forms a parallelogram. We take a quadrilateral ABCD and join their mid-points to form a quadrilateral EFGH. We draw a diagonal DB dividing the quadrilateral into ΔADB and ΔDBC. The Midpoint Theorem states that the segment joining two sides of a triangle at the mid-points of those sides is parallel to the third side and is half the length of the third side. Thus, HE || DB and HE = ½ DB.

Likewise, GF || DB and GF= ½ DB

Similarly when we draw the other diagonal AC, EF and HG || AC and EF = HG = ½ AC.

So we see that HE || GF and EF || HG. Thus, they form a parallelogram EFGH.

• Joining the Mid-Points of a Rhombus forms a Rectangle and vice versa.

We know that both rhombus and rectangle are different types of parallelograms.  So, when we join the mid points of the four sides of a rhombus we get a rectangle. And likewise when we join the mid-points of a rectangle, we get a rhombus.

Properties based on angles

• Bisectors of Alternate Angles on Parallel lines form a Rectangle.

AB and CD are two parallel lines and EF is a transversal intersecting them. By the property of alternate angles, CFE and BEF will be equal to each other. By the same property, AEF and DFE will also be equal to each other. We know that angles on a straight line are supplementary. Thus, CFE + EFD = 180° and AEF + BEF = 180°. Each of these angles is bisected. The bisectors of AEF and CFE meet at a point G whereas the bisectors of FEB and EFD meet at a point H, thus forming GEF and FHE respectively. On bisecting these angles, the sum of the angles on the straight line will also be halved. This means that GFH and GEF = 90°. Similarly, EGF and EHF are also equal to 90°. Thus the figure EFGH is a rectangle.

• Bisectors of Interior Angles of a Parallelogram enclose a Rectangle.

To illustrate this we take a parallelogram ABCD, and bisect its interior angles DAB, ADC, ABC and BCD. We know that adjacent angles of a parallelogram are supplementary. Thus, DAB and ADC add up to 180°. The bisectors of DAB and ADC meet at a point E, thus forming ΔADE. On bisecting them, the sum of their angles will be halved, i.e. 90°.  The sum of angles of a triangle is 180°, so the DEA will be 90°. This angle is vertically opposite to the interior angle of the figure enclosed by the bisectors. Similarly, the other interior angles of the figure will also be 90°. And as we have studied earlier, a quadrilateral whose 90° is a rectangle. Thus, the figure enclosed by the Bisector of interior angles of a parallelogram is a rectangle.

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