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- Quadrilaterals : Key Points About Quadrilaterals : Exercise

**Exercise 1**

**Question: 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 = 360Â°

Hence, angles are: 36Â°, 60Â°, 108Â°, 156Â°

**Question: 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.

**Question: 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.

**Question: 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.

**Question: 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.

**Question: 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

**Question: 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

**Question: 8. ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD. Show that**

**Question: 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

1. Sum of the angles of a quadrilateral is 360Â°.

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.

Understanding Quadrilaterals will be available online in PDF book form soon. The solutions are absolutely Free. Soon you will be able to download the solutions.