Understanding Quadrilaterals

Exercise 1

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

So, AB=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,

If DO=AO

(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

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 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.