SELINA Solutions for Class 9 Maths Chapter 9 - Triangles [Congruency in Triangles]

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Chapter 9 - Triangles [Congruency in Triangles] Exercise Ex. 9(A)

Question 1

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

(a) In Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesABC and Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesDEF, AB = DE, BC = EF and Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesB = Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesE.

(b) In Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesABC and Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesDEF, Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesB = Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesE = 90o; AC = DF and BC = EF.

(c) In Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesABC and Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesQRP, AB = QR, Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesB = Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesR and Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesC = P.

(d) In Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesABC and Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesPQR, AB = PQ, AC = PR and BC = QR.

(e) In Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesADC and Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesPQR, BC = QR, Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesA = 90o, Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesC = Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesR = 40o and Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesQ = 50o.

Solution 1

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles



Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles


Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles



Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles



Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles




Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Question 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

Question 3

The following figure shows a circle with centre O.

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

Solution 3

Question 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

Question 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

Question 6

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

Prove that: IA = IB = IC.

Solution 6

Question 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

Question 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

Question 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

Question 10

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

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

If Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesABD = 58o,

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesDBC = (2x - 4)o,

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesACB = y + 15o and

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesDCB = 63o ; find the values of x and y.

Solution 10

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Question 11

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

(i) BG = DF 

(ii) CF = EG

 Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles 

Solution 11

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

 

Question 12

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

Solution 12

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles 

Question 13

In the following figure, BL = CM.

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles 

 

Prove that AD is a median of triangle ABC.

Solution 13

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles 

Question 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

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles 

Solution 14

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles 

Question 15

Use the information in the given figure to prove :

(i) AB = FE

(ii) BD = CF 

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles 

Solution 15

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles 

Chapter 9 - Triangles [Congruency in Triangles] Exercise Ex. 9(B)

Question 1

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

Prove that: (i) Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesCAD = Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesBAE    (ii) CD = BE.

Solution 1

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

 

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

 

 

Question 2

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

In each case,

(i) Prove that: Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesAPD Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesSelina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesBPC

(ii) Find the angles of Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesDPC.

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

 

 

 

 

 

Solution 2

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

 

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

 

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Question 3

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

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesACQ and Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesASB are congruent.

(ii) CQ = BS.

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Solution 3

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesSelina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesSelina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Question 4

In a Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesABC, 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

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

 

 

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Question 5

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

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

If XS Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesQR and XT Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesPQ ; prove that:

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

(ii) PX bisects angle P.

Solution 5

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Question 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

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Question 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: Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesBEC Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesSelina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesDCF.

Solution 7

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Question 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 Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesABC and Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesPQR are congruent.

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Solution 8

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Question 9

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

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Prove that:

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

(ii) AB and PQ bisect each other.

Solution 9

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

Question 10

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

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Prove that:

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesAOB = 90o

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesAOD Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesSelina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesCOD

(iii) AD = CD

 

Solution 10

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Question 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) Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesAMCSelina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesSelina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesANB

(iii) BN = CM    (iv) Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesBMC Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesSelina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In TrianglesCNB

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Solution 11

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Question 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

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles 

Question 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

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles 

Question 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.

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles 

Prove that QR bisects PC.

 

Hint: (Show that QBP is equilateral

BP = PQ, but BP = CR

PQ = CR QPM RCM) 

Solution 14

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles 

Question 15

In the following figure, A = C and AB = BC. Prove that ΔABD ΔCBE. 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles 

Solution 15

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles 

 

 

In triangles AOE and COD,

A = C  (given)

AOE = COD  (vertically opposite angles)  

A + AOE = C + COD

180° - AEO = 180° - CDO

AEO = CDO  ….(i)

Now, AEO + OEB = 180° (linear pair)

And, CDO + ODB = 180° (linear pair)

AEO + OEB = CDO + ODB

OEB = ODB [Using (i)]

CEB = ADB ….(ii)

Now, in ΔABD and ΔCBE,

A = C  (given)

ADB = CEB  [From (ii)]

AB = BC  (given)

ΔABD ΔCBE  (by AAS congruence criterion) 

Question 16

AD and BC are equal perpendiculars to a line segment AB. If AD and BC are on different sides of AB prove that CD bisects AB.

Solution 16

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles 

 

 

In ΔAOD and ΔBOC,

AOD = BOC (vertically opposite angles)

DAO = CBO (each 90°)

AD = BC (given)

ΔAOD ΔBOC (by AAS congruence criterion) 

AO = BO [cpct]

O is the mid-point of AB.

Hence, CD bisects AB.

Question 17

In ΔABC, AB = AC and the bisectors of angles B and C intersect at point O. Prove that :

(i) BO = CO

(ii) AO bisects angle BAC.

Solution 17

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles 

 

 

In ΔABC,

AB = AC

B = C (angles opposite to equal sides are equal)

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Question 18

In the following figure, AB = EF, BC = DE and B = E = 90°.

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles 

 

Prove that AD = FC.

Solution 18

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles 

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

Question 19

A point O is taken inside a rhombus ABCD such that its distance from the vertices B and D are equal. Show that AOC is a straight line.

Solution 19

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles 

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles

 

Question 20

In quadrilateral ABCD, AD = BC and BD = CA. Prove that:

(i) ADB = BCA

(ii) DAB = CBA

Solution 20

Selina Solutions Icse Class 9 Mathematics Chapter - Triangles Congruency In Triangles 

In ΔABD and ΔBAC,

AD = BC (given)

BD = CA (given)

AB = AB (common)

ΔABD ΔBAC (by SSS congruence criterion)

 ADB = BCA and DAB = CBA (cpct)