Chapter 12 : Congruent Triangles - Rd Sharma Solutions for Class 9 Maths CBSE

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Chapter 12 - Congruent Triangles Excercise Ex. 12.1

Question 1

In fig., the sides BA and CA have been produced such that BA = AD and CA = AE.

 

Solution 1

Question 2

Solution 2



Question 3

Prove that the medians of an equilateral triangle are equal.

Solution 3



Question 4

Solution 4



Question 5

Solution 5



Question 6

The vertical angle of an isosceles triangle is 100o. Find its base angles.

Solution 6

Question 7

 In fig., AB = Ac and ACD = 105°, find BAC. 

 

Solution 7

Question 8

Solution 8



Question 9

Solution 9



Question 10

In fig., AB =AC and DB = DC, find the ratio ABD = ACD. 

 

Solution 10

Question 11

Determine the measure of each of the equal angles of a right-angled isosceles triangle.

                                                                                   OR

ABC is a right-angled triangle in which A = 90o and AB = AC. Find B and C.

Solution 11

Question 12

 


Solution 12

Question 13

AB is a line segment. P and Q are points on opposite sides of AB such that each of them is equidistant from the points A and B (See fig.). Show that the line PQ is perpendicular bisector of AB.

Solution 13

Chapter 12 - Congruent Triangles Excercise Ex. 12.2

Question 1

Solution 1



Question 2

In fig., it is given RT = TS, 1 = 22 and 4 = 23 prove that ΔRBT  ΔSAT.

 

Solution 2

Question 3

Solution 3



Chapter 12 - Congruent Triangles Excercise Ex. 12.3

Question 1

In two right triangles one side and acute angle of one are equal to the corresponding side and angle of the other. Prove that the triangles are congruent.

Solution 1





Let ABC and DEF be two right triangles.

Question 2

Solution 2



Question 3

Solution 3



Question 4
Show that the angles of an equilateral triangle are 60o each.
Solution 4
 
Let us consider that ABC is an equilateral triangle.
So, AB = BC = AC
Now, AB = AC
 C = B         (angles opposite to equal sides of a triangle are equal)
    
We also have
AC = BC    
 B = A             (angles opposite to equal sides of a triangle are equal)
    
So, we have
A = B = C
    Now, in ΔABC
A + B + C = 180o
 A + A + A = 180o
 3A = 180o
 A = 60o
 A = B = C = 60o
Hence, in an equilateral triangle all interior angles are of 60o.

Question 5

Solution 5



Question 6

Solution 6



Question 7

Solution 7



Question 8

Solution 8



Question 9

Solution 9



Question 10

Solution 10


Chapter 12 - Congruent Triangles Excercise Ex. 12.4

Question 1

In fig., it is given that Ab = CD and AD = BC. prove that ΔADC  ΔCBA

 

 

Solution 1

Question 2

Solution 2

Chapter 12 - Congruent Triangles Excercise Ex. 12.5

Question 1

Solution 1



Question 2

Solution 2



Question 3

Solution 3



Question 4

In fig., AD ⊥ CD and CB  CD. If AQ = BP an DP = CQ, prove that DAQ = CBP.

 

Solution 4

Question 5

Which of the following statements are True (T) and which are False (f):

(i) Sides opposite to equal angles of a triangle may be unequal.

(ii) Angles opposite to equal sides of a triangle are equal.

(iii) The measure of each angle of an equilaterial triangle is 60o.

(iv) If the altitude from one vertex of a triangle bisects the opposite side, then the triangle may be isoscles.

(v) The bisectors of two equal angles of a traingle are equal.

(vi) If the bisector of the vertical angle of a triangle bisects the base, then the triangle may be isosceles.

(vii) The two altitudes corresponding to two equal sides of a triangle need not be equal.

(viii) If any two sides of a right triangle are respectively equal to two sides of other right triagnle, then the two triangles are congruent.

(ix) Two right triangles are congruent if hypotenuse and a side of one triangle are respectively equal to the hypotenuse and a side of the other triangle.

Solution 5

(i) False

(ii) True

(iii) True

(iv) False

(v) True

(vi) False

(vii) False

(viii) False

(ix) True

Question 6

Solution 6

(i) equal

(ii) equal

(iii) equal

(iv) BC

(v) AC

(vi) equal to

(vii) EFD

Question 7

Solution 7



Chapter 12 - Congruent Triangles Excercise Ex. 12.6

Question 1

Solution 1



Question 2

Solution 2



Question 3

Solution 3



Question 4

Is it possible to draw a triangle with sides of length 2cm, 3cm and 7 cm?

Solution 4

Here, 2 + 3 < 7

Hence, it is not possible because triangle can be drawn only if the sum of any two sides is greater than third side.

Question 5

Solution 5



Question 6

Solution 6



Question 7

In fig., prove that:

i. CD + DA + AB + BC > 2AC

ii. CD + DA + AB > BC

 

Solution 7

Question 8

Which of the following statements are true (T) and which are false (F)?

(i) Sum of the three sides of a triangle is less than the sum of its three altitudes.

(ii) Sum of any two sides of a triangle is greater than twice the median drawn to the third side.

(iii) Sum of any two sides of a triangle is greater than the third side.

(iv) Difference of any two sides of a triangle is equal to the third side.

(v) If two angles of a triangle are unequal, then the greater angle has the larger side opposite to it.

(vi) Of all the line segments that can be drawn from a point to a line not containing it, the perpendicular line segment is the shortest one.

Solution 8

(i) False

(ii) True

(iii) True

(iv) False

(v) True

(vi) True

Question 9

Solution 9

(i) largest

(ii) less

(iii) greater

(iv) smaller

(v) less

(vi) greater

Question 10

Solution 10

Question 11

Solution 11



Chapter 12 - Congruent Triangles Excercise 12.85

Question 1

begin mathsize 12px style If space triangle ABC space approximately equal to space triangle LKM comma space then space side space of space triangle LKM space equal space to space side space AC space of space triangle ABC space is
left parenthesis straight a right parenthesis space LK
left parenthesis straight b right parenthesis space KM
left parenthesis straight c right parenthesis thin space LM
left parenthesis straight d right parenthesis space None space of space these end style

Solution 1

begin mathsize 12px style If space triangle ABC space approximately equal to space triangle LKM comma space then space from space figure space AC space equals space LM.
Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 2

begin mathsize 12px style If space triangle ABC space approximately equal to space triangle ACB comma space then space triangle ABC space is space isosceles space with
left parenthesis straight a right parenthesis space AB space equals space AC
left parenthesis straight b right parenthesis space AB space equals space BC
left parenthesis straight c right parenthesis space AC space equals space BC
left parenthesis straight d right parenthesis space None space of space these end style

Solution 2

begin mathsize 12px style triangle ABC space approximately equal to space triangle ACB
rightwards double arrow AB space equals space AC
space space space space space space or
space space space space space space AC space equals space AB
So comma space in space triangle ABC space is space isosceles space with space AB space equals space AC.
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 3

begin mathsize 12px style If space triangle ABC space approximately equal to space triangle PQR space and space triangle ABC space is space not space congruent space to space triangle RPQ comma space then space which space of space the space following space is space not space true colon
left parenthesis straight a right parenthesis space BC space equals space PQ
left parenthesis straight b right parenthesis space AC space equals space PR
left parenthesis straight c right parenthesis space AB space equals space PQ
left parenthesis straight d right parenthesis space QR space equals space BC end style

Solution 3

begin mathsize 12px style triangle ABC space approximately equal to space triangle PQR
rightwards double arrow AB space equals space PR comma space AC space equals space PR comma space BC space equals space QR
triangle ABC space ≇ space triangle RQP
rightwards double arrow AB space not equal to space QR comma space AC space not equal to space RP comma space BC space not equal to space PQ
So comma space option space left parenthesis straight a right parenthesis space is space not space true.
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 4

begin mathsize 12px style In space triangles space ABC space and space PQR space three space equality space relations space between space some space parts space are space as space follows space colon
AB space equals space QP comma space angle straight B space equals space angle straight P space and space BC space equals space PR
State space which space of space the space congruence space conditions space applies space colon
left parenthesis straight a right parenthesis space SAS
left parenthesis straight b right parenthesis space ASA
left parenthesis straight c right parenthesis space SSS
left parenthesis straight d right parenthesis space RHS
end style

Solution 4

begin mathsize 12px style From space given space conditions comma space we space have
AB space equals space PQ
BC space equals space PR
And space the space angle space between space these space sides space are space also space equal
straight i. straight e. space angle straight B space equals space angle straight P
So space SAS space property.
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 5

begin mathsize 12px style In space triangles space ABC space and space PQR comma space if space angle straight A space equals space angle straight R comma space angle straight B space equals space angle straight P space and space AB space equals space RP comma space then space which space one space of space the
following space congruence space conditions space applies space colon
left parenthesis straight a right parenthesis space SAS
left parenthesis straight b right parenthesis space ASA
left parenthesis straight c right parenthesis space SSS
left parenthesis straight d right parenthesis space RHS end style

Solution 5

begin mathsize 12px style From space given space conditions comma
angle straight B space equals space angle straight P
angle straight A space equals space angle straight R
And space the space side space containing space then space is space also space equal
straight i. straight e. space AB space equals space PR
So space ASA space property.
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 6

begin mathsize 12px style If space triangle PQR space approximately equal to space triangle EFD comma space then space ED space equals
left parenthesis straight a right parenthesis space PQ
left parenthesis straight b right parenthesis space QR
left parenthesis straight c right parenthesis space PR
left parenthesis straight d right parenthesis space None space of space these end style

Solution 6

begin mathsize 12px style increment PQR space approximately equal to space increment EFD comma
rightwards double arrow space ED space equals space PR space left parenthesis congruent space side s space of space congruent space triangles right parenthesis
Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 7

begin mathsize 12px style If space triangle PQR space approximately equal to space triangle EFD comma space then space angle straight E space equals
left parenthesis straight a right parenthesis space angle straight P
left parenthesis straight b right parenthesis space angle straight Q
left parenthesis straight c right parenthesis space angle straight R
left parenthesis straight d right parenthesis space None space of space these end style

Solution 7

begin mathsize 12px style increment PQR space approximately equal to space increment EFD comma
rightwards double arrow angle straight E space equals space angle straight P space space space space space left parenthesis congruent space angles space of space congruent space triangles right parenthesis
Hence comma space c orrect space option space is space left parenthesis straight a right parenthesis. end style

Question 8

begin mathsize 12px style In space straight a space triangle ABC comma space if space AB space equals space AC space and space BC space is space produced space to space straight D space such space that space angle ACD space equals space 100 degree comma space then space angle straight A space equals
left parenthesis straight a right parenthesis space 20 degree
left parenthesis straight b right parenthesis space 40 degree
left parenthesis straight c right parenthesis space 60 degree
left parenthesis straight d right parenthesis space 80 degree end style

Solution 8

begin mathsize 12px style AB space equals space AC
rightwards double arrow angle ABC space equals space angle ACB space space space space left parenthesis Isoscles space triangle space Property right parenthesis
angle ACB space equals space 180 degree space minus space 100 degree equals space 80 degree
rightwards double arrow angle ABC space equals space angle ACB space equals space 80 degree
angle straight A space equals space 180 degree space minus space angle ACB space minus space angle ABC space equals space 180 degree space minus space 80 degree space minus space 80 degree equals space 20 degree
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 9

In an isosceles triangle, if the vertex angle is twice the sum of the base angles, then the measure of vertex angle of the triangle is

(a) 100°

(b) 120°

(c) 110°

(d) 130°

Solution 9

begin mathsize 12px style Let space triangle ABC space be space an space isosceles space triangle space with space
vertex space angle space equals space angle straight A space and space base space angles space equals space angle straight B space and space angle straight C
Now comma space angle straight A space equals space 2 left parenthesis angle straight B space plus space angle straight C right parenthesis
rightwards double arrow fraction numerator angle straight A over denominator 2 end fraction equals angle straight B plus angle straight C space space space space space space space space space space space.... open parentheses 1 close parentheses
Also space in space triangle ABC comma
angle straight A space plus space angle straight B space plus space angle straight C space equals space 180 degree
rightwards double arrow angle straight A space plus space left parenthesis angle straight B space plus space angle straight C right parenthesis space equals space 180
rightwards double arrow angle straight A space plus space fraction numerator angle straight A over denominator 2 end fraction space equals space 180 degree space space space space space space space space space.... left square bracket From space left parenthesis 1 right parenthesis right square bracket
rightwards double arrow 3 over 2 angle straight A space equals space 180 degree
rightwards double arrow angle straight A equals fraction numerator 180 degree cross times 2 over denominator 3 end fraction
rightwards double arrow angle straight A space equals space 120 degree
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 10

Which of the following is not a criterion for congruence of triangles?

(a) SAS

(b) SSA

(c) ASA

(d) SSS

Solution 10

begin mathsize 12px style If space two space triangles space have space two space congruent space sides space and space straight a space congruent space non minus included space angle comma
then space triangle straight s space are space not space necessarily space congruent. space This space is space why space there space is space no space apostrophe Side space Side space angle apostrophe
straight i. straight e. space SSA space postulate.
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Chapter 12 - Congruent Triangles Excercise 12.86

Question 1

begin mathsize 12px style In space the space figure comma space the space measure space of space angle straight B apostrophe straight A apostrophe straight C apostrophe space is
left parenthesis straight a right parenthesis space 50 degree
left parenthesis straight b right parenthesis space 60 degree
left parenthesis straight c right parenthesis space 70 degree
left parenthesis straight d right parenthesis space 80 degree
end style

Solution 1

begin mathsize 12px style In space triangle ABC space and space triangle straight A apostrophe straight B apostrophe straight C apostrophe comma
AB space equals space straight A apostrophe straight B apostrophe
BC space equals space straight B apostrophe straight C apostrophe
angle ABC space equals space angle straight A apostrophe straight B apostrophe straight C apostrophe
So space triangle ABC space approximately equal to space triangle straight A apostrophe straight B apostrophe straight C apostrophe space by space SAS space creterion
rightwards double arrow angle BAC space equals space angle straight B apostrophe straight A apostrophe straight C apostrophe
rightwards double arrow 3 straight x space equals space 2 straight x space plus space 20
rightwards double arrow straight x space equals space 20 degree
rightwards double arrow 2 straight x plus 20 equals 2 space cross times space 20 space plus space 20 space equals space 60 degree equals angle straight B apostrophe straight A apostrophe straight C apostrophe
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 2

begin mathsize 12px style If space ABC space and space DEF space are space two space triangles space such space that space triangle ABC space approximately equal to space triangle FDE space and space AB space equals space 5 space cm comma space angle straight B space equals space 40 degree
and space angle straight A space equals space 80 degree. space Then comma space which space of space the space following space is space true ?
left parenthesis straight a right parenthesis space DF space equals space 5 space cm comma space space angle straight F space equals space 60 degree
left parenthesis straight b right parenthesis space DE space equals space 5 space cm comma space angle straight E space equals space 60 degree
left parenthesis straight c right parenthesis space DF space equals space 5 space cm comma space angle straight E space equals space 60 degree
left parenthesis straight d right parenthesis thin space DE space equals space 5 space cm comma space angle straight D space equals space 40 degree end style

Solution 2

begin mathsize 12px style In space increment ABC comma
angle straight C equals 180 degree minus angle straight A plus angle straight B equals 180 degree minus 80 degree minus 40 degree equals 60 degree
triangle ABC space approximately equal to space triangle FDE
rightwards double arrow AB space equals space FD space equals space 5 space cm
rightwards double arrow angle straight B space equals space angle straight D space equals space 40 degree
rightwards double arrow angle straight A space equals space angle straight F space equals space 80 degree
rightwards double arrow angle straight C space equals space angle straight E space equals space 60 degree
rightwards double arrow DF equals FD space equals space 5 cm space and space angle straight E space equals space 60 degree
Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 3

begin mathsize 12px style In space the space figure comma space AB space perpendicular space BE space and space FE space perpendicular space BE. space space If space BC space equals space DE space and space AB space equals space EF comma space then space triangle ABD space is space congruent space to
left parenthesis straight a right parenthesis space triangle EFC
left parenthesis straight b right parenthesis thin space triangle ECF
left parenthesis straight c right parenthesis space triangle CEF
left parenthesis straight d right parenthesis space triangle FEC end style

Solution 3

begin mathsize 12px style AB space equals space EF
BC space equals space DE
BC space plus space CD space equals space DE space plus space CD space space left parenthesis adding space CD space both space sides right parenthesis
BC space plus space CD space equals space BD comma space space DE space plus space CD space equals space CE
So space BD space equals space CE
Now space Consider space triangle ABD comma space & space triangle FEC
AB space equals space FE
BD space equals space EC
angle ABD space equals space angle FEC space equals space 90 degree
So space triangle ABD space approximately equal to space triangle FEC space by space SAS space creterion.
Hence comma space correct space option space is space left parenthesis straight d right parenthesis. end style

Question 4

begin mathsize 12px style In space figure comma space if space AE space parallel to space DC space and space AB equals AC comma space the space value space of space angle ABD space is
left parenthesis straight a right parenthesis space 70 degree
left parenthesis straight b right parenthesis space 110 degree
left parenthesis straight c right parenthesis space 120 degree
left parenthesis straight d right parenthesis space 130 degree end style

Solution 4

begin mathsize 12px style If space AE space parallel to space DC space and space AC space is space transversal comma
then space angle FAC space equals space 70 degree space space left parenthesis Opposite space angles right parenthesis
Also space angle FAC space equals space angle ACB space equals space 70 degree space left parenthesis Alternate space angles right parenthesis
Since space AB space equals space AC comma space triangle ABC space is space isosceles.
So space angle ABC space equals space angle ACB
rightwards double arrow angle ABC space equals space 70 degree
Now space angle ABD space equals space 180 degree space minus space angle ABC space equals space 180 degree space minus space 70 degree equals space 110 degree
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Chapter 12 - Congruent Triangles Excercise 12.87

Question 1

In the figure, ABC is an isosceles triangle whose side AC is produced to E. Through C, CD is drawn parallel to BA. The value of x is

(a) 52°

(b) 76°

(c) 156°

(d) 104°

 

Solution 1

begin mathsize 12px style triangle ABC space is space isosceles
angle ABC space equals space angle ACB space equals space 52 degree
then space angle BAC space equals space 180 degree space minus space 52 degree space minus space 52 degree space equals space 76 degree
If space AB space parallel to space CD comma space AC space is space transversal
then space angle BAC space equals space angle ACD space left parenthesis Alternate space angles right parenthesis
rightwards double arrow angle ACD space equals space 76 degree space space space
Now space from space figure comma
angle ACD space plus space straight x degree space equals space 180 degree
rightwards double arrow straight x degree space equals space 180 degree space minus space 76 degree
rightwards double arrow straight x degree space equals space 104 degree
Hence comma space correct space option space is space left parenthesis straight d right parenthesis. end style

Question 2

begin mathsize 12px style In space figure comma space if space AC space is space bisector space of space angle BAD space such space that space AB space equals space 3 space cm space and space AC space equals space 5 space cm comma space then space CD space equals
left parenthesis straight a right parenthesis space 2 space cm
left parenthesis straight b right parenthesis space 3 space cm
left parenthesis straight c right parenthesis space 4 space cm
left parenthesis straight d right parenthesis space 5 space cm end style

Solution 2

begin mathsize 12px style Consider space triangle ABC space and space triangle ADC
angle ABC space equals space angle ADC space equals space 90 degree
angle BAC space equals space angle CAD space space space space space left parenthesis AC space is space bisector space of space angle straight A right parenthesis
Also space if space two space angles space are space equal comma space then space the space third space angle space will space also space be space equal.
rightwards double arrow angle BCA space equals space angle DCA
Now comma space AC space equals space AC space space space space left parenthesis common right parenthesis
So space by space ASA space property comma space triangle ABC space approximately equal to space triangle ADC
rightwards double arrow BC space equals space CD
And comma space BC space equals space square root of AC squared minus AB squared end root equals space square root of 25 space minus space 9 end root space equals space 4 space cm
rightwards double arrow CD space equals space 4 space cm
Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 3

begin mathsize 12px style straight D comma space straight E comma space straight F space are space the space mid minus points space of space the space sides space BC comma space CA space and space AB space respectively space of space triangle ABC. space Then space triangle DEF
is space congruent space to space triangle
left parenthesis straight a right parenthesis space ABC
left parenthesis straight b right parenthesis space AEF
left parenthesis straight c right parenthesis space BFD comma space CDE
left parenthesis straight d right parenthesis space AFE comma space FBD comma space EDC end style

Solution 3

begin mathsize 12px style In space any space triangle comma space straight a space line space joining space the space mid minus points space of space any space two space sides space is space parallel space to space the space third space side.
rightwards double arrow EF space parallel to space BC space space EF space parallel to DC space and space BD
Similiarly space DF space parallel to AC. space space space
rightwards double arrow DF space parallel to AE space and space EC
Also space DE space parallel to space AB. space space space
rightwards double arrow DE space parallel to space AF space space and space BF
From space this space information space it space is space clear space that space EFDC comma space EFBD comma space EAFD
are space the space parallelogram space by space property.
Now space consider space one space parallelogram space EFDC
Consider space triangle DEF space and space triangle EDC
DE space equals space ED space space space left parenthesis Common right parenthesis
angle DEF space equals space angle EDC
angle EDF space equals space angle DEC space space left parenthesis ASA space property right parenthesis
rightwards double arrow triangle DEF space approximately equal to space triangle EDC
Similiarly space in space Parallelogram space EAFD comma
triangle DEF space approximately equal to space triangle AFE
And space in space Parallelogram space EFBD
triangle DEF space approximately equal to space triangle FBD
Hence comma space corect space option space is space left parenthesis straight d right parenthesis.
Note colon space Option space left parenthesis straight d right parenthesis space modified. end style

Question 4

begin mathsize 12px style ABC space is space an space isosceles space triangle space such space that space AB space equals AC space and space AD space is space the space medium space to space base space BC. space Then comma space angle BAD space equals
left parenthesis straight a right parenthesis space 55 degree
left parenthesis straight b right parenthesis space 70 degree
left parenthesis straight c right parenthesis space 35 degree
left parenthesis straight d right parenthesis space 110 degree end style

Solution 4

begin mathsize 12px style If space AD space is space the space median comma space then space straight D space is space the space mid minus point space of space space BC.
rightwards double arrow BD space equals space DC
So space consider space triangle ADB space and space triangle ADC
AD space equals space AD space space left parenthesis Common right parenthesis
DB space equals space DC
BA space equals space CA
So space by space SSS comma space triangle ADB space approximately equal to space triangle ADC
Now space angle straight B space equals space angle straight C space equals space 35 degree
rightwards double arrow angle BAD space equals space angle DAC
So space in space triangle ABC comma
angle straight A space plus space angle straight B space plus space angle straight C space equals space 180 degree
rightwards double arrow 2 angle BAD space plus space 35 degree space plus space 35 degree space equals space 180 degree space
rightwards double arrow 2 angle BAD space equals space 110 degree space
rightwards double arrow angle BAD space equals space 55 degree
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 5

In figure, X is a point in the interior of square ABCD. AXYZ is also a square. If DY = 3 cm and AZ = 2 cm, then BY =

(a) 5 cm

(b) 6 cm

(c) 7 cm

(d) 8 cm

Solution 5

begin mathsize 12px style Consider space triangle AZD space and space triangle AXB
AZ space equals space AX space equals space 2 space cm space left parenthesis AXYZ space is space straight a space square right parenthesis
angle AZD space equals space angle AXB space equals space 90 degree
AD space equals space AB space left parenthesis ABCD space is space straight a space square right parenthesis
So space by space RHS space creterion comma space triangle AZD space approximately equal to space triangle AXB
rightwards double arrow ZD space equals space XB
Now comma space ZD space equals space ZY space plus space DY space space space space space space space space space space space space space space space space space space space space space space space
space space space space space space space space space space space space space space space space space space equals space 2 space cm space plus space 3 space cm space space space space left parenthesis ZY space equals space AZ space equals space 2 space cm right parenthesis
space space space space space space space space space space space space space space space space space space equals space 5 space cm
rightwards double arrow space XB space equals space 5 space cm space
rightwards double arrow space BY space equals space YX space plus space XB space space equals space 2 space cm space plus space 5 space cm space equals space 7 space cm
Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Chapter 12 - Congruent Triangles Excercise 12.88

Question 1

begin mathsize 12px style In space figure comma space ABC space is space straight a space triangle space in space which space angle straight B space equals space 2 angle straight C. space straight D space is space straight a space point space on space side space space BC space such space that space AD space space bisects space
angle BAC space and space AB space equals space CD. space BE space is space the space bisector space of space angle straight B. space The space measure space of space angle BAC space is
left parenthesis straight a right parenthesis space 72 degree
left parenthesis straight b right parenthesis space 73 degree
left parenthesis straight c right parenthesis space 74 degree
left parenthesis straight d right parenthesis space 95 degree end style

Solution 1

begin mathsize 12px style angle ABE space equals space angle EBC space left parenthesis BE space is space bisector space of space angle straight B right parenthesis
and space angle straight C space equals space fraction numerator angle straight B over denominator 2 end fraction
rightwards double arrow angle EBC space equals space angle ECB
So space triangle EBC space is space isosceles space triangle.
rightwards double arrow EB space equals space EC space space space.... open parentheses 1 close parentheses
Now space Consider space triangle ABE space and space triangle DCE
AB space equals space DC space space space space space left parenthesis Given right parenthesis
BE space equals space CE space space space space left square bracket From space left parenthesis 1 right parenthesis right square bracket
angle ABE space equals space angle DCE space left parenthesis From space above space data right parenthesis
So space triangle ABE space approximately equal to space triangle DCE space by space SAS space property
rightwards double arrow AE space equals space DE space space space
angle BAE space equals space angle CDE space equals space angle straight A
Now space consider space triangle AED comma space
AE space equals space DE space left parenthesis above space proved right parenthesis
rightwards double arrow triangle AED space is space isosceles space triangl
rightwards double arrow angle EAD space equals space angle EDA space equals space fraction numerator angle straight A over denominator 2 end fraction space space space space space space space left parenthesis AD space is space Bisector space of space angle straight A right parenthesis space space space.... open parentheses 2 close parentheses
Now comma space consider space triangle ABC comma
angle straight A space plus space angle straight B space plus space straight C space equals space 180 degree
rightwards double arrow angle straight A space plus space 2 angle straight C space plus space angle straight C space equals space 180 degree space space space space space space space space left parenthesis angle straight B space equals space 2 angle straight C right parenthesis
rightwards double arrow angle straight A space plus space 3 angle straight C space equals space 180 degree space space space space space space space space.... open parentheses 3 close parentheses
Consider space triangle ADC comma
rightwards double arrow fraction numerator angle straight A over denominator 2 end fraction plus angle ADC space plus space angle straight C space equals space 180 degree
rightwards double arrow fraction numerator angle straight A over denominator 2 end fraction plus left parenthesis angle EDA space plus space angle CDE right parenthesis space plus space angle straight C space equals space 180 degree
rightwards double arrow fraction numerator angle straight A over denominator 2 end fraction plus fraction numerator angle straight A over denominator 2 end fraction plus angle straight A space plus space angle straight C space equals space 180 degree space space space space
rightwards double arrow angle straight A plus angle straight A space plus space angle straight C space equals space 180 degree
rightwards double arrow 2 angle straight A space plus space angle straight C space equals space 180 degree space space space space space space space space space space space space space space space space space space space space space space space space space space.... open parentheses 4 close parentheses
Right space hand space side space of space equations space open parentheses 3 close parentheses space and space open parentheses 4 close parentheses space are space equal comma space hence space Left space hand space side.
rightwards double arrow angle straight A space plus space 3 angle straight C space equals space 2 angle straight A space plus space angle straight C
rightwards double arrow angle straight A space equals space 2 angle straight C
Substituting space in space equation space open parentheses 3 close parentheses comma
2 angle straight C space plus space 3 angle straight C space equals space 180 degree
rightwards double arrow 5 angle straight C equals 180 degree
rightwards double arrow angle straight C equals 36 degree
rightwards double arrow angle straight A space equals space 2 cross times 36 degree equals 72 degree
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

CBSE Class 9 Maths Homework Help

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