ML Aggarwal Class 9 Solutions Chapter 13 provides comprehensive guidance and step-by-step explanations for the concepts covered in the 13th chapter of the NCERT textbook for Class 9 Mathematics. This chapter typically introduces fundamental mathematical concepts, laying the groundwork for future studies.

## ML Aggarwal Class 9 Chapter 13 Solutions

### ICSE Class 9 Maths Chapter 13 Solutions ML Aggarwal

**Exercise 13.1**

**Question 1.**

**If two angles of a quadrilateral are 40° and 110° and the other two are in the ratio 3 : 4, find these angles.**

**Solution:**

**Question 2.**

**If the angles of a quadrilateral, taken in order, are in the ratio 1 : 2 : 3 : 4, prove that it is a trapezium.**

**Solution:**

**Question 3.**

**If an angle of a parallelogram is two-thirds of its adjacent angle, find the angles of the parallelogram.**

**Solution:**

**Question 4.**

**(a) In figure (1) given below, ABCD is a parallelogram in which ∠DAB = 70°, ∠DBC = 80°. Calculate angles CDB and ADB.**

**(b) In figure (2) given below, ABCD is a parallelogram. Find the angles of the AAOD.**

**(c) In figure (3) given below, ABCD is a rhombus. Find the value of x.**

**Solution:**

**Question 5.**

**(a) In figure (1) given below, ABCD is a parallelogram with perimeter 40. Find the values of x and y.**

**(b) In figure (2) given below. ABCD is a parallelogram. Find the values of x and y.**

**(c) In figure (3) given below. ABCD is a rhombus. Find x and y.**

**Solution:**

**Question 6.**

The diagonals AC and BD of a rectangle > ABCD intersect each other at P. If ∠ABD = 50°, find ∠DPC.

**Solution:**

**Question 7.**

**(a) In figure (1) given below, equilateral triangle EBC surmounts square ABCD. Find angle BED represented by x.**

**(b) In figure (2) given below, ABCD is a rectangle and diagonals intersect at O. AC is produced to E. If ∠ECD = 146°, find the angles of the ∆ AOB.**

**(c) In figure (3) given below, ABCD is rhombus and diagonals intersect at O. If ∠OAB : ∠OBA = 3:2, find the angles of the ∆ AOD.**

**Solution:**

**Question 8.**

**(a) In figure (1) given below, ABCD is a trapezium. Find the values of x and y.**

**(b) In figure (2) given below, ABCD is an isosceles trapezium. Find the values of x and.y.**

**(c) In figure (3) given below, ABCD is a kite and diagonals intersect at O. If ∠DAB = 112° and ∠DCB = 64°, find ∠ODC and ∠OBA.**

**Solution:**

**Question 9.**

**(i) Prove that each angle of a rectangle is 90°.**

**(ii) If the angle of a quadrilateral are equal, prove that it is a rectangle.**

**(iii) If the diagonals of a rhombus are equal, prove that it is a square.**

**(iv) Prove that every diagonal of a rhombus bisects the angles at the vertices.**

**Solution:**

**Question 10.**

**ABCD is a parallelogram. If the diagonal AC bisects ∠A, then prove that:**

**(i) AC bisects ∠C**

**(ii) ABCD is a rhombus**

**(iii) AC ⊥ BD.**

**Solution:**

**Question 11.**

**(i) Prove that bisectors of any two adjacent angles of a parallelogram are at right angles.**

**(ii) Prove that bisectors of any two opposite angles of a parallelogram are parallel.**

**(iii) If the diagonals of a quadrilateral are equal and bisect each other at right angles, then prove that it is a square.**

**Solution:**

**Question 12.**

**(i) If ABCD is a rectangle in which the diagonal BD bisect ∠B, then show that ABCD is a square.**

**(ii) Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.**

**Solution:**

**Question 13.**

**P and Q are points on opposite sides AD and BC of a parallelogram ABCD such that PQ passes through the point of intersection O of its diagonals AC and BD. Show that PQ is bisected at O.**

**Solution:**

**Question 14.**

**(a) In figure (1) given below, ABCD is a parallelogram and X is mid-point of BC. The line AX produced meets DC produced at Q. The parallelogram ABPQ is completed. Prove that:**

**(i) the triangles ABX and QCX are congruent;**

**(ii)DC = CQ = QP**

**(b) In figure (2) given below, points P and Q have been taken on opposite sides AB and CD respectively of a parallelogram ABCD such that AP = CQ. Show that AC and PQ bisect each other.**

**Solution:**

**Question 15.**

**ABCD is a square. A is joined to a point P on BC and D is joined to a point Q on AB. If AP=DQ, prove that AP and DQ are perpendicular to each other.**

**Solution:**

**Question 16.**

**If P and Q are points of trisection of the diagonal BD of a parallelogram ABCD, prove that CQ || AP.**

**Solution:**

**Question 17.**

**A transversal cuts two parallel lines at A and B. The two interior angles at A are bisected and so are the two interior angles at B ; the four bisectors form a quadrilateral ABCD. Prove that**

**(i) ABCD is a rectangle.**

**(ii) CD is parallel to the original parallel lines.**

**Solution:**

**Question 18.**

**In a parallelogram ABCD, the bisector of ∠A meets DC in E and AB = 2 AD. Prove that**

**(i) BE bisects ∠B**

**(ii) ∠AEB = a right angle.**

**Solution:**

**Question 19.**

**ABCD is a parallelogram, bisectors of angles A and B meet at E which lie on DC. Prove that AB**

**Solution:**

**Question 20.**

**ABCD is a square and the diagonals intersect at O. If P is a point on AB such that AO =AP, prove that 3 ∠POB = ∠AOP.**

**Solution:**

**Question 21.**

**ABCD is a square. E, F, G and H are points on the sides AB, BC, CD and DA respectively such that AE = BF = CG = DH. Prove that EFGH is a square**.

**Solution:**

**Question 22.**

**(a) In the Figure (1) given below, ABCD and ABEF are parallelograms. Prove that**

**(i) CDFE is a parallelogram**

**(ii) FD = EC**

**(iii) Δ AFD = ΔBEC.**

**(b) In the figure (2) given below, ABCD is a parallelogram, ADEF and AGHB are two squares. Prove that FG = AC**

**Solution:**

**Question 23.**

**ABCD is a rhombus in which ∠A = 60°. Find the ratio AC : BD.**

**Solution:**

**Exercise 13.2**

**Question 1.**

**Using ruler and compasses only, construct the quadrilateral ABCD in which ∠ BAD = 45°, AD = AB = 6cm, BC = 3.6cm, CD = 5cm. Measure ∠ BCD.**

**Solution:**

**Question 2.**

**Draw a quadrilateral ABCD with AB = 6cm, BC = 4cm, CD = 4 cm and ∠ ABC = ∠ BCD = 90°**

**Solution:**

**Question 3.**

**Using ruler and compasses only, construct the quadrilateral ABCD given that AB = 5 cm, BC = 2.5 cm, CD = 6 cm, ∠BAD = 90° and the diagonal AC = 5.5 cm.**

**Solution:**

**Question 4.**

**Construct a quadrilateral ABCD in which AB = 3.3 cm, BC = 4.9 cm, CD = 5.8 cm, DA = 4 cm and BD = 5.3 cm.**

**Solution:**

**Question 5.**

**Construct a trapezium ABCD in which AD || BC, AB = CD = 3 cm, BC = 5.2cm and AD = 4 cm**

**Solution:**

**Question 6.**

**Construct a trapezium ABCD in which AD || BC, ∠B= 60°, AB = 5 cm. BC = 6.2 cm and CD = 4.8 cm.**

**Solution:**

**Question 7.**

**Using ruler and compasses only, construct a parallelogram ABCD with AB = 5.1 cm, BC = 7 cm and ∠ABC = 75°.**

**Solution:**

**Question 8.**

**Using ruler and compasses only, construct a parallelogram ABCD in which AB = 4.6 cm, BC = 3.2 cm and AC = 6.1 cm.**

**Solution:**

**Question 9.**

**Using ruler and compasses, construct a parallelogram ABCD give that AB = 4 cm, AC = 10 cm, BD = 6 cm. Measure BC.**

**Solution:**

**Question 10.**

**Using ruler and compasses only, construct a parallelogram ABCD such that BC = 4 cm, diagonal AC = 8.6 cm and diagonal BD = 4.4 cm. Measure the side AB.**

**Solution:**

**Question 11.**

**Use ruler and compasses to construct a parallelogram with diagonals 6 cm and 8 cm in length having given the acute angle between them is 60°. Measure one of the longer sides.**

**Solution:**

**Question 12.**

**Using ruler and compasses only, draw a parallelogram whose diagonals are 4 cm and 6 cm long and contain an angle of 75°. Measure and write down the length of one of the shorter sides of the parallelogram.**

**Solution:**

**Question 13.**

**Using ruler and compasses only, construct a parallelogram ABCD with AB = 6 cm, altitude = 3.5 cm and side BC = 4 cm. Measure the acute angles of the parallelogram.**

**Solution:**

**Question 14.**

**The perpendicular distances between the pairs of opposite sides of a parallelogram ABCD are 3 cm and 4 cm and one of its angles measures 60°. Using ruler and compasses only, construct ABCD.**

**Solution:**

**Question 15.**

**Using ruler and compasses, construct a rectangle ABCD with AB = 5cm and AD = 3 cm.**

**Solution:**

**Question 16.**

**Using ruler and compasses only, construct a rectangle each of whose diagonals measures 6cm and the diagonals intersect at an angle of 45°.**

**Solution:**

**Question 17.**

**Using ruler and compasses only, construct a square having a diagonal of length 5cm. Measure its sides correct to the nearest millimeter.**

**Solution:**

**Question 18.**

**Using ruler and compasses only construct A rhombus ABCD given that AB 5cm, AC = 6cm measure ∠BAD.**

**Solution:**

**Question 19.**

**Using ruler and compasses only, construct rhombus ABCD with sides of length 4cm and diagonal AC of length 5 cm. Measure ∠ABC.**

**Solution:**

**Question 20.**

**Construct a rhombus PQRS whose diagonals PR and QS are 8cip and 6cm respectively.**

**Solution:**

**Question 21.**

**Construct a rhombus ABCD of side 4.6 cm and ∠BCD = 135°, by using ruler and compasses only.**

**Solution:**

**Question 22.**

**Construct a trapezium in which AB || CD, AB = 4.6 cm, ∠ ABC = 90°, ∠ DAB = 120° and the distance between parallel sides is 2.9 cm.**

**Solution:**

**Question 23.**

**Construct a trapezium ABCD when one of parallel sides AB = 4.8 cm, height = 2.6cm, BC = 3.1 cm and AD = 3.6 cm.**

**Solution:**

**Question 24.**

**Construct a regular hexagon of side 2.5 cm.**

**Solution:**

**Multiple Choice Questions**

**Choose the correct answer from the given four options (1 to 12):**

**Question 1.**

**Three angles of a quadrilateral are 75°, 90° and 75°. The fourth angle is**

**(a) 90°**

**(b) 95°**

**(c) 105°**

**(d) 120°**

**Solution:**

Sum of 4 angles of a quadrilateral = 360° Sum of three angles = 75° + 90° + 75° = 240° Fourth angle = 360° – 240° = 120° **(d)**

**Question 2.**

**A quadrilateral ABCD is a trapezium if**

**(a) AB = DC**

**(b) AD = BC**

**(c) ∠A + ∠C = 180°**

**(d) ∠B + ∠C = 180°**

**Solution:**

A quadrilateral ABCD is a trapezium if ∠B + ∠C= 180°

(Sum of co-interior angles) **(d)**

**Question 3.**

**If PQRS is a parallelogram, then ∠Q – ∠S is equal to**

**(a) 90°**

**(b) 120°**

**(c) 0°**

**(d) 180°**

**Solution:**

PQRS is a parallelogram ∠Q – ∠S = 0

(∵ Opposite angles of a parallelogram, are equal)** (c)**

**Question 4.**

**A diagonal of a rectangle is inclined to one side of the rectangle at 25°. The acute angle between the diagonals is**

**(a) 55°**

**(b) 50°**

**(c) 40°**

**(d) 25°**

**Solution:**

In a rectangle a diagonal is inclined to one side of the rectangle is 25°

**Question 5.**

**ABCD is a rhombus such that ∠ACB = 40°. Then ∠ADB is**

**(a) 40°**

**(b) 45°**

**(c) 50°**

**(d) 60°**

**Solution:**

**Question 6.**

**The diagonals AC and BD of a parallelogram ABCD intersect each other at the point O. If ∠D AC = 32° and ∠AOB = 70°, then ∠DBC is equal to**

**(a) 24°**

**(b) 86°**

**(c) 38°**

**(d) 32°**

**Solution:**

**Question 7.**

**If the diagonals of a square ABCD intersect each other at O, then ∆OAB is**

**(a) an equilateral triangle**

**(b) a right angled but not an isosceles triangle**

**(c) an isosceles but not right angled triangle**

**(d) an isosceles right angled triangle**

**Solution:**

**Question 8.**

**If the diagonals of a quadrilateral PQRS bisect each other, then the quadrilateral PQRS must be a**

**(a) parallelogram**

**(b) rhombus**

**(c) rectangle**

**(d) square**

**Solution:**

Diagonals of a quadrilateral PQRS bisect each other, then quadrilateral must be a parallelogram.

(∵ A rhombus, rectangle and square are also parallelogram)** (a)**

**Question 9.**

**If the diagonals of a quadrilateral PQRS bisect each other at right angles, then the quadrilateral PQRS must be a**

**(a) parallelogram**

**(b) rectangle**

**(c) rhombus**

**(d) square**

**Solution:**

Diagonals of quadrilateral PQRS bisect each other at right angles, then quadrilateral PQRS [ must be a rhombus.

(∵ Square is also a rhombus with each angle equal to 90°)** (c)**

**Question 10.**

**Which of the following statement is true for a parallelogram?**

**(a) Its diagonals are equal.**

**(b) Its diagonals are perpendicular to each other.**

**(c) The diagonals divide the parallelogram into four congruent triangles.**

**(d) The diagonals bisect each other.**

**Solution:**

For a parallelogram an the statement ‘The diagoanls bisect each other’ is true.** (d)**

**Question 11.**

**Which of the following is not true for a parallelogram?**

**(a) opposite sides are equal**

**(b) opposite angles are equal**

**(c) opposite angles are bisected by the diagonals**

**(d) diagonals bisect each other**

**Solution:**

The statement that in a parallelogram, .the opposite angles are bisected by the diagonals, is not true in each case. **(c)**

**Question 12.**

**A quadrilateral in which the diagonals are equal and bisect each other at right angles is a**

**(a) rectangle which is not a square**

**(b) rhombus which is not a square**

**(c) kite which is not a square**

**(d) square**

**Solution:**

In a quadrilateral, if diagonals are equal and bisect each other at right angles, is a square. **(d)**

**Chapter Test**

**Question P.Q.**

**The interior angles of a polygon add upto 4320°. How many sides does the polygon have ?**

**Solution:**

**Question P.Q.**

**If the ratio of an interior angle to the exterior angle of a regular polygon is 5:1, find the number of sides.**

**Solution:**

**Question P.Q.**

**In a pentagon ABCDE, BC || ED and ∠B: ∠A : ∠E =3:4:5. Find ∠A.**

**Solution:**

**Question 1.**

**In the given figure, ABCD is a parallelogram. CB is produced to E such that BE=BC. Prove that AEBD is a parallelogram.**

**Solution:**

**Question 2.**

**In the given figure, ABC is an isosceles triangle in which AB=AC. AD bisects exterior angle PAC and CD || BA. Show that**

**(i) ∠DAC=∠BCA**

**(ii) ABCD is a parallelogram.**

**Solution:**

**Question 3.**

**Prove that the quadrilateral obtained by joining the mid-points of an isosceles trapezium is a rhombus.**

**Solution:**

**Question 4.**

**Find the size of each lettered angle in the Following Figures.**

**Solution:**

**Question 5.**

**Find the size of each lettered angle in the following figures :**

**Solution:**

**Question 6.**

**In the adjoining figure, ABCD is a rhombus and DCFE is a square. If ∠ABC = 56°, find**

**(i) ∠DAG**

**(ii) ∠FEG**

**(iii) ∠GAC**

**(iv) ∠AGC.**

**Solution:**

**Question 7.**

**If one angle of a rhombus is 60° and the length of a side is 8 cm, find the lengths of its diagonals.**

**Solution:**

**Question 8.**

**Using ruler and compasses only, construct a parallelogram ABCD with AB = 5 cm, AD = 2.5 cm and ∠BAD = 45°. If the bisector of ∠BAD meets DC at E, prove that ∠AEB is a right angle.**

**Solution:**