# Triangles ML Aggarwal ICSE 9th Maths Solutions

**Triangles ML Aggarwal ICSE 9th** Maths Solutions APC.Step by Step Answer of Exercise-10.1, Exercise-10.2, Exercise-10.3, Exercise-10.4 MCQ and Chapter-Test Questions of **Triangles**** **for **ICSE** **Class-9** Mathematics**.** So This post is the Answer of **ML Aggarwal Triangles** Exercise Questions for **ICSE** Maths Class-9**.** Visit official website CISCE for detail information about ICSE Board Class-9th.

**Triangles ML Aggarwal ICSE 9th Maths Solutions Chapter-10**

**–: Select Topics :–**

Note:- Before viewing Solution of **triangle **Chapter of **ML Aggarwal ICSE Class-9** Maths clear the next point**. **Angle sum property and concurrency of** triangle** is the base of this chapter but inequality in **Triangle** is also important properties in this chapter.Type of **triangle** on the base of **angle** and on the base of side should be clear before solving this chapter . Incentre and Incircle topic with circumcentre and circumcircle are main concept of **Triangle**.

### Exercise 10.1 Triangles** ML Aggarwal Solutions **

**Question 1.**

**It is given that ∆ABC ≅ ∆RPQ. Is it true to say that BC = QR ? Why?**

**Answer 1**

**Question 2.**

**“If two sides and an angle of one triangle are equal to two sides and an angle of another triangle, then the two triangles must be congruent.” Is the statement true? Why?**

**Answer 2**

No, it is not true statement as the angles should be included angle of there two given sides.

**Question 3.**

**In the given figure, AB=AC and AP=AQ. Prove that**

**(i) ∆APC ≅ ∆AQB**

**(ii) CP = BQ**

**(iii) ∠APC = ∠AQB.**

**Answer 3**

**Question 4.**

**In the given figure, AB = AC, P and Q are points on BA and CA respectively such that AP = AQ. Prove that**

**(i) ∆APC ≅ ∆AQB**

**(ii) CP = BQ**

**(iii) ∠ACP = ∠ABQ.**

**Answer 4**

**Question 5.**

**In the given figure, AD = BC and BD = AC. Prove that :**

**∠ADB = ∠BCA and ∠DAB = ∠CBA.**

**Answer 5 **

**Question 6.**

**In the given figure, ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA. Prove that**

**(i) ∆ABD ≅ ∆BAC**

**(ii) BD = AC**

**(iii) ∠ABD = ∠BAC.**

**Answer 6**

**Question 7.**

**In the given figure, AB = DC and AB || DC. Prove that AD = BC.**

**Answer 7**

**Question 8**

**In the given figure. AC = AE, AB = AD and ∠BAD = ∠CAE. Show that BC = DE.**

**Answer 8**

**Question 9.**

**In the adjoining figure, AB = CD, CE = BF and ∠ACE = ∠DBF. Prove that**

**(i) ∆ACE ≅ ∆DBF**

**(ii) AE = DF.**

**Answer 9**

**Question 10.**

**In the given figure, AB = AC and D is mid-point of BC. Use SSS rule of congruency to show that**

**(i) ∆ABD ≅ ∆ACD**

**(ii) AD is bisector of ∠A**

**(iii) AD is perpendicular to BC.**

**Answer 10**

**Question 11.**

**Two line segments AB and CD bisect each other at O. Prove that :**

**(i) AC = BD**

**(ii) ∠CAB = ∠ABD**

**(iii) AD || CB**

**(iv) AD = CB.**

**Answer 11**

**Question 12.**

**In each of the following diagrams, find the values of x and y.**

**Answer 12**

** Triangle Answer of Exercise 10.2 for Class-9 Understanding**

**Question 1.**

**In triangles ABC and PQR, ∠A= ∠Q and ∠B = ∠R. Which side of APQR should be equal to side AB of AABC so that the two triangles are congruent? Give reason for your answer.**

**Answer 1**

**Question 2.**

**In triangles ABC and PQR, ∠A = ∠Q and ∠B = ∠R. Which side of APQR should be equal to side BC of AABC so that the two triangles are congruent? Give reason for your answer.**

**Answer 2**

**Question 3.**

**“If two angles and a side of one triangle are equal to two angles and a side of another triangle, then the two triangles must be congruent”. Is the statement true? Why?**

**Answer 3**

The given statement can be true only if the corresponding (included) sides are equal otherwise not.

**Question 4.**

**In the given figure, AD is median of ∆ABC, BM and CN are perpendiculars drawn from B and C respectively on AD and AD produced. Prove that BM = CN.**

**Answer 4**

**Question 5.**

**In the given figure, BM and DN are perpendiculars to the line segment AC. If BM = DN, prove that AC bisects BD.**

**Answer 5**

**Question 6.**

**In the given figure, l and m are two parallel lines intersected by another pair of parallel lines p and q. Show that ∆ABC ≅ ∆CDA.**

**Answer 6**

**Question 7.**

**In the given figure, two lines AB and CD intersect each other at the point O such that BC || DA and BC = DA. Show that O is the mid-point of both the line segments AB and CD.**

**Answer 7**

**Question 8.**

**In the given figure, ∠BCD = ∠ADC and ∠BCA = ∠ADB. Show that**

**(i) ∆ACD ≅ ∆BDC**

**(ii) BC = AD**

**(iii) ∠A = ∠B.**

**Answer 8**

**Question 9.**

**In the given figure, ∠ABC = ∠ACB, D and E are points on the sides AC and AB respectively such that BE = CD. Prove that**

**(i) ∆EBC ≅ ∆DCB**

**(ii) ∆OEB ≅ ∆ODC**

**(iii) OB = OC.**

**Answer 9**

**Question 10.**

**ABC is an isosceles triangle with AB=AC. Draw AP ⊥ BC to show that ∠B = ∠C.**

**Answer 10**

**Question 11.**

**In the given figure, BA ⊥ AC, DE⊥ DF such that BA = DE and BF = EC.**

**Answer 11**

**Question 12.**

**ABCD is a rectanige. X and Y are points on sides AD and BC respectively such that AY = BX. Prove that BY = AX and ∠BAY = ∠ABX.**

**Answer 12**

**Question 13.**

**(a) In the figure (1) given below, QX, RX are bisectors of angles PQR and PRQ respectively of A PQR. If XS⊥ QR and XT ⊥ PQ, prove that**

**(i) ∆XTQ ≅ ∆XSQ**

**(ii) PX bisects the angle P.**

**(b) In the figure (2) given below, AB || DC and ∠C = ∠D. Prove that**

**(i) AD = BC**

**(ii) AC = BD.**

**(c) In the figure (3) given below, BA || DF and CA II EG and BD = EC . Prove that, .**

**(i) BG = DF**

**(ii) EG = CF.**

**Answer 13**

**Question 14.**

**In each of the following diagrams, find the values of x and y.**

**Answer 14**

**Exercise-10.3 ML Aggarwal Class-9**

**Question 1.**

**ABC is a right angled triangle in which ∠A = 90° and AB = AC. Find ∠B and ∠C.**

**Answer 1**

**Question 2.**

**Show that the angles of an equilateral triangle are 60° each.**

**Answer 2**

**Question 3.**

**Show that every equiangular triangle is equilateral.**

**Answer 3**

**Question 4.**

**In the following diagrams, find the value of x:**

**Answer 4**

**Question 5.**

**In the following diagrams, find the value of x:**

**Answer 5**

**Question 6.**

**(a) In the figure (1) given below, AB = AD, BC = DC. Find ∠ ABC.**

**(b)In the figure (2) given below, BC = CD. Find ∠ACB.**

**(c) In the figure (3) given below, AB || CD and CA = CE. If ∠ACE = 74° and ∠BAE =15°, find the values of x and y.**

**Answer 6**

**Question 7.**

**In ∆ABC, AB = AC, ∠A = (5x + 20)° and each of the base angle is th of ∠A. Find the measure of ∠A.**

**Answer 7**

**Question 8.**

**(a) In the figure (1) given below, ABC is an equilateral triangle. Base BC is produced to E, such that BC’= CE. Calculate ∠ACE and ∠AEC.**

**(b) In the figure (2) given below, prove that ∠ BAD : ∠ ADB = 3 : 1.**

**(c) In the figure (3) given below, AB || CD. Find the values of x, y and ∠.**

**Answer 8**

**Question 9.**

**In the given figure, D is mid-point of BC, DE and DF are perpendiculars to AB and AC respectively such that DE = DF. Prove that ABC is an isosceles triangle.**

**Answer 9**

**Question 10.**

**In the given figure, AD, BE and CF arc altitudes of ∆ABC. If AD = BE = CF, prove that ABC is an equilateral triangle.**

**Answer 10**

**Question 11.**

**In a triangle ABC, AB = AC, D and E are points on the sides AB and AC respectively such that BD = CE. Show that:**

**(i) ∆DBC ≅ ∆ECB**

**(ii) ∠DCB = ∠EBC**

**(iii) OB = OC,where O is the point of intersection of BE and CD.**

**Answer 11**

**Question 12.**

**ABC is an isosceles triangle in which AB = AC. P is any point in the interior of ∆ABC such that ∠ABP = ∠ACP. Prove that**

**(a) BP = CP**

**(b) AP bisects ∠BAC.**

**Answer 12**

**Question 13.**

**In the adjoining figure, D and E are points on the side BC of ∆ABC such that BD = EC and AD = AE. Show that ∆ABD ≅ ∆ACE.**

**Answer 13**

**Question 14.**

**(a) In the figure (i) given below, CDE is an equilateral triangle formed on a side CD of a square ABCD. Show that ∆ADE ≅ ∆BCE and hence, AEB is an isosceles triangle.**

**(b) In the figure (ii) given below, O is a point in the interior of a square ABCD such that OAB is an equilateral trianlge. Show that OCD is an isosceles triangle.**

**Answer 14**

**Question 15.**

**In the given figure, ABC is a right triangle with AB = AC. Bisector of ∠A meets BC at D. Prove that BC = 2AD.**

**Answer 15**

**Triangle Exercise-10.4 Answer of ML Aggarwal ICSE Class-9**

**Question 1.**

**In ∆PQR, ∠P = 70° and ∠R = 30°. Which side of this triangle is longest? Give reason for your answer.**

**Answer 1**

**Question 2.**

**Show that in a right angled triangle, the hypotenuse is the longest side.**

**Answer 2**

**Question 3.**

**PQR is a right angle triangle at Q and PQ : QR = 3:2. Which is the least angle.**

**Answer 3**

**Question 4.**

**In ∆ ABC, AB = 8 cm, BC = 5.6 cm and CA = 6.5 cm. Which is (i) the greatest angle ?**

**(ii) the smallest angle ?**

**Answer 4**

**Question 5.**

**In ∆ABC, ∠A = 50°, ∠B= 60°, Arrange the sides of the triangle in ascending order**

**.Answer 5**

**Question 6.**

**In figure given alongside, ∠B = 30°, ∠C = 40° and the bisector of ∠A meets BC at D. Show**

**(i) BD > AD**

**(ii) DC > AD**

**(iii) AC > DC**

**(iv) AB > BD**

**Answer 6**

**Question 7.**

**(a) In the figure (1) given below, AD bisects ∠A. Arrange AB, BD and DC in the descending order of their lengths.**

**(b) In the figure (2) given below, ∠ ABD = 65°, ∠DAC = 22° and AD = BD. Calculate ∠ ACD and state (giving reasons) which is greater : BD or DC ?**

**Answer 7**

**Question 8.**

**(a) In the figure (1) given below, prove that (i) CF> AF (ii) DC>DF.**

**(b) In the figure (2) given below, AB = AC.**

**Prove that AB > CD.**

**(c) In the figure (3) given below, AC = CD. Prove that BC < CD.**

**Answer 8**

**Question 9.**

**(a) In the figure (i) given below, ∠B < ∠A and ∠C < ∠D. Show that AD < BC. (b) In the figure (ii) given below, D is any point on the side BC of ∆ABC. If AB > AC, show that AB > AD.**

**Answer 9**

**Question 10.**

**(i) Is it possible to construct a triangle with lengths of its sides as 4 cm, 3 cm and 7 cm? Give reason for your answer,**

**(ii) Is it possible to construct a triangle with lengths of its sides as 9 cm, 7 cm and 17 cm? Give reason for your answer.**

**(iii) Is it possible to construct a triangle with lengths of its sides as 8 cm, 7 cm and 4 cm? Give reason for your answer.**

**Answer 10**

(i) Length of sides of a triangle are 4 cm, 3 cm and 7 cm

We know that sum of any two sides of a triangle is greatar than its third side But 4 + 3 = 7 cm

Which is not possible

Hence to construction of a triangle with sides 4 cm, 3 cm and 7 cm is not possible.

(ii) Length of sides of a triangle are 9 cm, 7 cm and 17 cm

We know that sum of any two sides of a triangle is greater than its third side Now 9 + 7 = 16 < 17 ∴ It is not possible to construct a triangle with these sides.

(iii) Length of sides of a triangle are 8 cm, 7 cm and 4 cm We know that sum of any two sides of a triangle is greater than its third side Now 7 + 4 = 11 > 8

Yes, It is possible to construct a triangle with these sides.

**MCQ Triangle Answer Exercise 10 ML Aggarwal**

**Choose the correct answer from the given four options (1 to 18):**

**Question 1.**

**Which of the following is not a criterion for congruency of triangles?**

**(a) SAS**

**(b) ASA**

**(c) SSA**

**(d) SSS**

**Answer 1**

Criteria of congruency of two triangles ‘SSA’ is not the criterion**. (c)**

**Question 2.**

**In the adjoining figure, AB = FC, EF=BD and ∠AFE = ∠CBD. Then the rule by which ∆AFE = ∆CBD is**

**(a) SAS**

**(b) ASA**

**(c) SSS**

**(d) AAS**

**Answer 2**

**Question 3.**

**In the adjoining figure, AB ⊥ BE and FE ⊥ BE. If AB = FE and BC = DE, then**

**(a) ∆ABD ≅ ∆EFC**

**(b) ∆ABD ≅ ∆FEC**

**(c) ∆ABD ≅ ∆ECF**

**(d) ∆ABD ≅ ∆CEF**

**Answer 3**

In the figure given,

**Question 4.**

**Answer 4**

**Question 5.**

**In the adjoining figure, O is mid point of AB. If ∠ACO = ∠BDO, then ∠OAC is equal to**

**(a) ∠OCA**

**(b) ∠ODB**

**(c) ∠OBD**

**(d) ∠BOD **

**Answer 5**

**Question 6.**

**In the adjoining figure, AC = BD. If ∠CAB = ∠DBA, then ∠ACB is equal to**

**(a) ∠BAD**

**(b) ∠ABC**

**(c) ∠ABD**

**(d) ∠BDA**

**Answer 7**

**Question 7.**

**In the adjoining figure, ABCD is a quadrilateral in which BN and DM are drawn perpendiculars to AC such that BN = DM. If OB = 4 cm, then BD is**

**(a) 6 cm**

**(b) 8 cm**

**(c) 10 cm**

**(d) 12 cm**

**Answer 7**

**Question 8.**

**In ∆ABC, AB = AC and ∠B = 50°. Then ∠C is equal to**

**(a) 40°**

**(b) 50°**

**(c) 80°**

**(d) 130°**

**Answer 8**

**Question 9.**

**In ∆ABC, BC = AB and ∠B = 80°. Then ∠A is equal to**

**(a) 80°**

**(b) 40°**

**(c) 50°**

**(d) 100°**

**Answer 9**

**Question 10.**

**In ∆PQR, ∠R = ∠P, QR = 4 cm and PR = 5 cm. Then the length of PQ is**

**(a) 4 cm**

**(b) 5 cm**

**(c) 2 cm**

**(d) 2.5 cm**

**Answer 10**

**Question 11.**

**In ∆ABC and APQR, AB = AC, ∠C = ∠P and ∠B = ∠Q. The two triangles are**

**(a) isosceles but not congruent**

**(b) isosceles and congruent**

**(c) congruent but isosceles**

**(d) neither congruent nor isosceles**

**Answer 11**

**Question 12.**

**Two sides of a triangle are of lenghts 5 cm and 1.5 cm. The length of the third side of the triangle can not be**

**(a) 3.6 cm**

**(b) 4.1 cm**

**(c) 3.8 cm**

**(d) 3.4 cm**

**Answer 12**

**Question 13.**

**If a, b, c are the lengths of the sides of a trianlge, then**

**(a) a – b > c**

**(b) c > a + b**

**(c) c = a + b**

**(d) c < A + B**

**Answer 13**

a, b, c are the lengths of the sides of a trianlge than a + b> c or c < a + b

(Sum of any two sides is greater than its third side)** (d)**

**Question 14.**

**It is not possible to construct a triangle when the lengths of its sides are**

**(a) 6 cm, 7 cm, 8 cm**

**(b) 4 cm, 6 cm, 6 cm**

**(c) 5.3 cm, 2.2 cm, 3.1 cm**

**(d) 9.3 cm, 5.2 cm, 7.4 cm**

**Answer 14**

We know that sum of any two sides of a triangle is greater than its third side 2.2 + 3.1 = 5.3 ⇒ 5.3 = 5.3 is not possible (c)

**Question 15.**

**In ∆PQR, if ∠R> ∠Q, then**

**(a) QR > PR**

**(b) PQ > PR**

**(c) PQ < PR**

**(d) QR < PR**

**Answer 15**

In ∆PQR, ∠R> ∠Q

∴ PQ > PR **(b)**

**Question 16.**

**If triangle PQR is right angled at Q, then**

**(a) PR = PQ**

**(b) PR < PQ**

**(c) PR < QR
(d) PR > PQ**

**Answer 16**

**Question 17.**

**If triangle ABC is obtuse angled and ∠C is obtuse, then**

**(a) AB > BC**

**(b) AB = BC**

**(c) AB < BC**

**(d) AC > AB**

**Answer 17**

**Question 18.**

**If the lengths of two sides of an isosceles are 4 cm and 10 cm, then the length of the third side is**

**(a) 4 cm**

**(b) 10 cm**

**(c) 7 cm**

**(d) 14 cm**

**Answer 18**

Lengths of two sides of an isosceles triangle are 4 cm and 10 cm, then length of the third side is 10 cm

(Sum of any two sides of a triangle is greater than its third side and 4 cm is not possible as 4 + 4 > 10 cm.

** Chapter-Test of Triangle Class-9 for ML Aggarwal Answer**

**Question 1.**

**In triangles ABC and DEF, ∠A = ∠D, ∠B = ∠E and AB = EF. Will the two triangles be congruent? Give reasons for your answer.**

**Answer 1**

**Question 2.**

**In the adjoining figure, ABCD is a square. P, Q and R are points on the sides AB, BC and CD respectively such that AP= BQ = CR and ∠PQR = 90°. Prove that**

**(a) ∆PBQ ≅ ∆QCR**

**(b) PQ = QR**

**(c) ∠PRQ = 45°**

**Answer 2**

**Question 3.**

**In the given figure, OA ⊥ OD, OC X OB, OD = OA and OB = OC. Prove that AB = CD.**

**Answer 3**

**Question 4.**

**In the given figure, PQ || BA and RS CA. If BP = RC, prove that:**

**(i) ∆BSR ≅ ∆PQC**

**(ii) BS = PQ**

**(iii) RS = CQ.**

**Answer 4**

**Question 5.**

**In the given figure, AB = AC, D is a point in the interior of ∆ABC such that ∠DBC = ∠DCB. Prove that AD bisects ∠BAC of ∆ABC**.

**Answer 5**

**Question 6**

**In the adjoining figure, AB || DC. CE and DE bisects ∠BCD and ∠ADC respectively. Prove that AB = AD + BC.**

**Answer 6**

**Question 7**

**In ∆ABC, D is a point on BC such that AD is the bisector of ∠BAC. CE is drawn parallel to DA to meet BD produced at E. Prove that ∆CAE is isosceles**

**Answer 7**

**Question 8.**

**In the figure (ii) given below, ABC is a right angled triangle at B, ADEC and BCFG are squares. Prove that AF = BE.**

**Answer 8**

**Question 9**

**In the given figure, BD = AD = AC. If ∠ABD = 36°, find the value of x .**

**Answer 9**

**Question 10**

**In the adjoining figure, TR = TS, ∠1 = 2∠2 and ∠4 = 2∠3. Prove that RB = SA.**

**Answer 10**

**Given:** In the figure , RST is a triangle

**Question 11**

**(a) In the figure (1) given below, find the value of x.**

**(b) In the figure (2) given below, AB = AC and DE || BC. Calculate**

**(i)x**

**(ii) y**

**(iii) ∠BAC**

**(c) In the figure (1) given below, calculate the size of each lettered angle.**

**Answer 11**

**Question 12**

**(a) In the figure (1) given below, AD = BD = DC and ∠ACD = 35°. Show that**

**(i) AC > DC (ii) AB > AD.**

**(b) In the figure (2) given below, prove that**

**(i) x + y = 90° (ii) z = 90° (iii) AB = BC**

**Answer 12**

**Question 13.**

**In the adjoining figure, ABC and DBC are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC. If AD is extended to intersect BC at P, show that**

**(i) ∆ABD ≅ ∆ACD**

**(ii) ∆ABP ≅ ∆ACP**

**(iii) AP bisects ∠A as well as ∠D**

**(iv) AP is the perpendicular bisector of BC.**

**Answer 13**

**Question 14**

**In the given figure, AP ⊥ l and PR > PQ. Show that AR > AQ.**

**Answer 14**

**Question 15**

**If O is any point in the interior of a triangle ABC, show that**

**OA + OB + OC > **

**(AB + BC + CA).**

**Answer 15**