1. The angles of quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the quadrilateral.
Answer: As you know angle sum of a quadrilateral = 360°
Hence, angles are: 36°, 60°, 108°, 156°
2. If the diagonals of a parallelogram are equal, then show that it is a rectangle.
Answer: In the following parallelogram both diagonals are equal:
As all are right angles so the parallelogram is a rectangle.
3. Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.
Answer: In the given quadrilateral ABCD diagonals AC and BD bisect each other at right angle. We have to prove that AB=BC=CD=AD
Similarly AB=BC=CD=AD can be proved which means that ABCD is a rhombus.
4. Show that the diagonals of a square are equal and bisect each other at right angles.
Answer: In the figure given above let us assume that
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.
5. Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.
Answer: Using the same figure,
(Angles opposite to equal sides are equal)
So, all angles of the quadrilateral are right angles making it a square.
6. Diagonal AC of a parallelogram ABCD bisects angle A . Show that
(i) it bisects angle C also,
(ii) ABCD is a rhombus.
Answer: ABCD is a parallelogram where diagonal AC bisects angle DAB
As diagonals are intersecting at right angles so it is a rhombus
7. In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ. Show that:
With equal opposite angles and equal opposite sides it is proved that APCQ is a parallelogram
8. ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD. Show that
9. In ∆ ABC and ∆ DEF, AB = DE, AB || DE, BC = EF and BC || EF. Vertices A, B and C are joined to vertices D, E and F respectively. Show that
(i) quadrilateral ABED is a parallelogram
(ii) quadrilateral BEFC is a parallelogram
(iii) AD || CF and AD = CF
(iv) quadrilateral ACFD is a parallelogram
(v) AC = DF
In quadrilateral ABED
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
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 Quadrilaterals
1. 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.