Selina Concise Mathematics Class 9 ICSE Solutions Chapter 9 - Triangles [Congruency in Triangles]

Selina Concise Mathematics Class 9 ICSE Solutions Triangles [Congruency in Triangles]

Selina ICSE Solutions for Class 9 Maths Chapter 9 Triangles [Congruency in Triangles]

Exercise 9(A)


1.Which of the following pairs of triangles are congruent? In each case, state the condition of congruency:

(a) In ABC and DEF, AB = DE, BC = EF and B = E.

(b) In ABC and DEF, B = E = 90^o; AC = DF and BC = EF.

(c) In ABC and QRP, AB = QR, B = R and C = P.

(d) In ABC and PQR, AB = PQ, AC = PR and BC = QR.

(e) In ADC and PQR, BC = QR, A = 90^o, C = R = 40^o and Q = 50^o.

Solution 1:

2.The given figure shows a circle with centre O. P is mid-point of chord AB.




Show that OP is perpendicular to AB.

Solution 2:

3.The following figure shows a circle with centre O.


If OP is perpendicular to AB, prove that AP = BP.

Solution 3:

4.In a triangle ABC, D is mid-point of BC; AD is produced upto E so that DE = AD. Prove that:

(i) ABD and ECD are congruent.

(ii) AB = CE.

(iii) AB is parallel to EC.

Solution 4:

5.A triangle ABC has B = C.

Prove that:

(i) The perpendiculars from the mid-point of BC to AB and AC are equal.

(ii) The perpendiculars form B and C to the opposite sides are equal.

Solution 5:

6.The perpendicular bisectors of the sides of a triangle ABC meet at I.

Prove that: IA = IB = IC.

Solution 6:

7.A line segment AB is bisected at point P and through point P another line segment PQ, which is perpendicular to AB, is drawn. Show that: QA = QB.

Solution 7:

8.If AP bisects angle BAC and M is any point on AP, prove that the perpendiculars drawn from M to AB and AC are equal.

Solution 8:

9.From the given diagram, in which ABCD is a parallelogram, ABL is al line segment and E is mid point of BC.

Prove that:

(i) DCE  LBE

(ii) AB = BL.

(iii) AL = 2DC

Solution 9:

10.In the given figure, AB = DB and Ac = DC.

If ABD = 58^o,

DBC = (2x - 4)^o,

ACB = y + 15^o and

DCB = 63^o ; find the values of x and y.

Solution 10:

11.In the given figure: AB//FD, AC//GE and BD = CE; prove that:

(i) BG = DF 

(ii) CF = EG

  

Solution 11:

12.In ∆ABC, AB = AC. Show that the altitude AD is median also.

Solution 12:

13.In the following figure, BL = CM.

 

Prove that AD is a median of triangle ABC.

Solution 13:

14.In the following figure, AB = AC and AD is perpendicular to BC. BE bisects angle B and EF is perpendicular to AB.

Prove that :

(i) BD = CD

(ii) ED = EF

Solution 14:

15.Use the information in the given figure to prove :

(i) AB = FE

(ii) BD = CF 




Solution 15:

16.

Solution 16:

Exercise 9(B)

1.On the sides AB and AC of triangle ABC, equilateral triangle ABD and ACE are drawn.

Prove that: (i) CAD = BAE    (ii) CD = BE.

Solution 1:

2.In the following diagrams, ABCD is a square and APB is an equilateral triangle.

In each case,

(i) Prove that: APD BPC

(ii) Find the angles of DPC.

Solution 2:

3.In the figure, given below, triangle ABC is right-angled at B. ABPQ and ACRS are squares. Prove that:

(i) ACQ and ASB are congruent.

(ii) CQ = BS.

Solution 3:

4.In a ABC, BD is the median to the side AC, BD is produced to E such that BD = DE. Prove that: AE is parallel to BC.

Solution 4:

5.In the adjoining figure, OX and RX are the bisectors of the angles Q and R respectively of the triangle PQR.

If XS QR and XT PQ ; prove that:

(i) 

(ii) PX bisects angle P.

Solution 5:

6.In the parallelogram ABCD, the angles A and C are obtuse. Points X and Y are taken on the diagonal BD such that the angles XAD and YCB are right angles.

Prove that: XA = YC.

Solution 6:

7.ABCD is a parallelogram. The sides AB and AD are produced to E and F respectively, such produced to E and F respectively, such that AB = BE and AD = DF.

Prove that:  BEC DCF.

Solution 7:

8.In the following figures, the sides AB and BC and the median AD of triangle ABC are equal to the sides PQ and QR and median PS of the triangle PQR. Prove that ABC and PQR are congruent.

Solution 8:

9.In the following diagram, AP and BQ are equal and parallel to each other.

Prove that:

(i) 

(ii) AB and PQ bisect each other.

Solution 9:

10.In the following figure, OA = OC and AB = BC.

Prove that:

(i) AOB = 90^o

(ii) AOD COD

(iii) AD = CD

Solution 10:

11.The following figure shown a triangle ABC in which AB = AC. M is a point on AB and N is a point on AC such that BM = CN.

Prove that:

(i) AM = AN      (ii) AMCANB

(iii) BN = CM    (iv) BMC CNB

Solution 11:

12.In a triangle ABC, AB = BC, AD is perpendicular to side BC and CE is perpendicular to side AB. Prove that : AD = CE.

Solution 12:

13.PQRS is a parallelogram. L and M are points on PQ and SR respectively such that PL= MR. Show that LM and QS bisect each other.

Solution 13:

14.In the following figure, ABC is an equilateral triangle in which QP is parallel to AC. Side AC is produced upto point R so that CR = BP.

 

Prove that QR bisects PC.

Hint: (Show that ∆ QBP is equilateral

⇒ BP = PQ, but BP = CR

⇒ PQ = CR ⇒ ∆ QPM ≅ ∆ RCM) 

Solution 14:

15.

Solution 15:

16.

Solution 16:

17.

Solution 17:

18.

Solution 18:

19.

Solution 19:

Selina Concise Mathematics Class 9 ICSE Solutions Chapter 9 - Triangles [Congruency in Triangles]