## Understanding Quadrilaterals
Hence, angles are: 36°, 60°, 108°, 156°
As all are right angles so the parallelogram is a rectangle.
So, AB=AD Similarly AB=BC=CD=AD can be proved which means that ABCD is a rhombus.
DO=AO (Sides opposite equal angles are equal) Similarly AO=OB=OC can be proved This gives the proof of diagonals of square being equal.
If DO=AO (Angles opposite to equal sides are equal) So, all angles of the quadrilateral are right angles making it a square.
As diagonals are intersecting at right angles so it is a rhombus
With equal opposite angles and equal opposite sides it is proved that APCQ is a parallelogram
In quadrilateral ABED AB= ED AB||ED So, ABED is a parallelogram (opposite sides are equal and parallel) So, BE||AD ------------ (1) Similarly quadrilateral ACFD can be proven to be a parallelogram So, BE||CF ------------ (2) From equations (1) & (2) It is proved that AD||CF So, AD=CF Similarly AC=DF and AC||DF can be proved 10. ABCD is a trapezium in which AB || CD and AD = BC. Show that ## Key Points About Quadrilaterals1. Sum of the angles of a quadrilateral is 360°. 2. A diagonal of a parallelogram divides it into two congruent triangles. 3. In a parallelogram, (i) opposite sides are equal (ii) opposite angles are equal (iii) diagonals bisect each other 4. A quadrilateral is a parallelogram, if (i) opposite sides are equal or (ii) opposite angles are equal or (iii) diagonals bisect each other or (iv) a pair of opposite sides is equal and parallel 5. Diagonals of a rectangle bisect each other and are equal and vice-versa. 6. Diagonals of a rhombus bisect each other at right angles and vice-versa. 7. Diagonals of a square bisect each other at right angles and are equal, and vice-versa. 8. The line-segment joining the mid-points of any two sides of a triangle is parallel to the third side and is half of it. 9. A line through the mid-point of a side of a triangle parallel to another side bisects the third side. 10. The quadrilateral formed by joining the mid-points of the sides of a quadrilateral, in order, is a parallelogram. |