Chapter 11 : Triangle and its Angles - Rd Sharma Solutions for Class 9 Maths CBSE

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Chapter 11 - Triangle and its Angles Excercise Ex. 11.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Two angles of a triangle are equal and the third angle is greater than each of those angles by 30o. Determine all the angles of the triangle.

Solution 4

Question 5

Solution 5

Question 6

Can a triangle have:

(i) Two right angles?

(ii) Two obtuse angles?

(iii) Two acute angles?

(iv) All angles more than 60o?

(v) All angles less than 60o?

(vi) All angles equal to 60o?

 

Justify your answer in each case.

Solution 6

(i) No

As two right angles would sum up to 180o, and we know that the sum of all three angles of a triangle is 180o, so the third angle will become zero. This is not possible, so a triangle cannot have two right angles.

(ii) No

A triangle cannot have 2 obtuse angles, since then the sum of those two angles will be greater than 180which is not possible as the sum of all three angles of a triangle is 180o.

(iii) Yes

A triangle can have 2 acute angles.

(iv) No

The sum of all the internal angles of a triangle is 180o. Having all angles more than 60o will make that sum more than 180o, which is impossible.

(v) No

The sum of all the internal angles of a triangle is 180o. Having all angles less than 60o will make that sum less than 180o, which is impossible.

(vi) Yes


The sum of all the internal angles of a triangle is 180o.  So, a triangle can have all angles as 60o. Such triangles are called equilateral triangles.

Question 7

begin mathsize 12px style The space angles space of space straight a space triangle space are space arranged space in space ascending space order space of space magnitude. space
If space the space difference space between space two space consecutive space angles space is space 10 to the power of 0 comma space find space the space three space angles. end style

Solution 7

begin mathsize 12px style Let space three space angles space be colon space open parentheses straight x minus 10 close parentheses to the power of 0 comma space space straight x space space and space space open parentheses straight x plus 10 close parentheses to the power of 0
therefore space straight x to the power of 0 plus space open parentheses straight x minus 10 close parentheses to the power of 0 plus space open parentheses straight x plus 10 close parentheses to the power of 0 space equals space 180 to the power of 0 space space space space space space space space end exponent open square brackets Sum space of space all space angles space of space straight a triangle space close square brackets
rightwards double arrow 3 straight x space equals space 180 to the power of 0
rightwards double arrow straight x space equals space 60 to the power of 0
Since space straight x space space equals space 60 to the power of 0
open parentheses straight x minus 10 to the power of 0 close parentheses space equals space 60 to the power of 0 minus 10 to the power of 0 equals 50 to the power of 0
open parentheses straight x plus 10 to the power of 0 close parentheses space equals space 60 to the power of 0 plus 10 to the power of 0 equals 70 to the power of 0
therefore space first space angle space equals space 50 to the power of 0
space space space space space space second space angle space equals space 60 to the power of 0
space space space space space third space angle space equals space 70 to the power of 0 end style

Question 8

Solution 8



Question 9

Solution 9



Question 10

Solution 10



Question 11

Solution 11




Question 12

Solution 12

Chapter 11 - Triangle and its Angles Excercise Ex. 11.2

Question 1

The exterior angles, obtained on producing both the base of a triangle both ways are 104o and 136o. Find all the angles of the triangle.

Solution 1



Question 2

 

In fig., the sides BC, CA and AB of a ΔABC have been produced to D, E, and F respectively. If ACD = 105° and EAF = 45°, find all the angles of the ΔABC.

 

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

In fig., AC  CE and A : B : C = 3 : 2 : 1, find the value of ECD.

 

 

Solution 6

Question 7

In fig. AB  DE. Find ACD.

 

Solution 7

Question 8

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

Solution 8

Question 9

Fill in the blanks to make the following statements true:

(i) Sum of the angle of triangle is ______ .

(ii) An exterior angle of a triangle is equal to the two ______ opposite angles.

(iii) An exterior angle of a traingle is always _______ than either of the interior oppsite angles.

(iv) A traingle cannot have more than ______ right angles.

(v) A triangles cannot have more than ______ obtuse angles.

Solution 9

(i) 180o

(ii) interior

(iii) greater

(iv) one

(v) one

Question 10

Solution 10



Question 11

 

Solution 11



Question 12

In fig., AB divides DAC in the ratio 1 : 3 and AB = DB. Determine the value of x.

 

 

Solution 12

Question 13



Solution 13



Question 14

In fig., AM  BC and AN is the bisector of A. If B = 65° and C = 33°, find MAN.

Solution 14

Question 15

Solution 15



Question 16

Solution 16



Question 17

In fig. AE bisects CAD and B = C. Prove that AE  BC.

 

 

Solution 17

Chapter 11 - Triangle and its Angles Excercise 11.25

Question 1

If all the three angles of a triangle are equal, then each one of them is equal to

(a) 90°

(b) 45°

(c) 60°

(d) 30°

Solution 1

Let the measure of each angle be x°.

Now, the sum of all angles of any triangle is 180°.

Thus, x° + x° + x° = 180°

i.e. 3x° = 180°

i.e. x° = 60°

Hence, correct option is (c).

Question 2

If two acute angles of a right triangle are equal, then each acute is equal to

(a) 30°

(b) 45°

(c) 60°

(d) 90°

Solution 2

Let the measure of each acute angle of a triangle be x°.

Then, we have

x° + x° + 90° = 180°

i.e. 2x° = 90°

i.e. x° = 45°

Hence, correct option is (b).

 

Question 3

An exterior angle of a triangle is equal to 100° and two interior opposite angle are equal. Each of these angles is equal to

(a) 75°

(b) 80°

(c) 40°

(d) 50°

Solution 3

 begin mathsize 12px style Let space the space two space interior space opposite space angles space be space straight x degree space each. space space
Now comma space the space exterior space angle space is space equal space to space the space sum space of space the space two space in t erior space opposite space angles.
straight x degree space plus space straight x degree space equals space 180 degree
rightwards double arrow 2 straight x degree space equals space 100 degree
rightwards double arrow straight x degree space equals space 50 degree
Hence comma space correct space option space is space left parenthesis straight d right parenthesis. space end style

Question 4

If one angle of a triangle is equal to the sum of the other two angles, then the triangle is

(a) an isosceles triangle

(b) an obtuse triangle

(c) an equilateral triangle

(d) a right triangle

Solution 4

Let the three angles of a triangle be A, B and C.

Now, A + B + C = 180°

If A = B + C

Then A + (A) = 180°

i.e. 2A = 180°

i.e. A = 90°

Since, one of the angle is 90°, the triangle is a Right triangle.

Hence, correct option is (d).

Question 5

begin mathsize 12px style Side space BC space of space straight a space triangle space ABC space has space been space produced space to space straight a space point space straight D space such space that space angle space ACD space equals space 120 degree.
If space angle space straight B space equals space 1 half space angle space straight A comma space then space angle space straight A space is space equal space to
left parenthesis straight a right parenthesis space 80 degree
left parenthesis straight b right parenthesis space 75 degree
left parenthesis straight c right parenthesis space 60 degree
left parenthesis straight d right parenthesis space 90 degree end style

Solution 5

begin mathsize 12px style angle straight B equals 1 half angle straight A
angle ACD space is space an space exterior space angle.
rightwards double arrow angle straight A plus angle straight B equals angle ACD
rightwards double arrow angle straight A plus 1 half angle straight A equals 120 degree
rightwards double arrow fraction numerator 3 angle straight A over denominator 2 end fraction equals 120 degree
rightwards double arrow 3 angle straight A equals 240 degree
rightwards double arrow angle straight A equals 80 degree
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 6

begin mathsize 12px style In space ΔABC comma space angle straight B space equals space angle straight C space and space ray space AX space bisects space the space exterior space angle space angle DAC. space If space angle DAX space equals space 70 degree comma space then space angle ACB space equals
left parenthesis straight a right parenthesis space 35 degree
left parenthesis straight b right parenthesis space 90 degree
left parenthesis straight c right parenthesis space 70 degree
left parenthesis straight d right parenthesis space 55 degree end style

Solution 6

begin mathsize 12px style AX space is space bisector space of space angle DAC.
rightwards double arrow angle DAX space equals space angle XAC space equals space 70 degree
rightwards double arrow angle DAC space equals space 2 space cross times space 70 space equals space 140 degree
Now comma space angle straight A space equals space 180 degree space minus space angle DAC space equals space 180 degree space minus space 140 degree space equals space 40 degree
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 40 degree space plus space angle straight B space plus space angle straight C space equals space 180 degree
rightwards double arrow 40 degree space plus angle straight C space plus space angle straight C space equals space 180 degree space space space space space space space.... left parenthesis angle straight B space equals space angle straight C right parenthesis
rightwards double arrow 2 angle straight C space equals space 140 degree
rightwards double arrow angle straight C space equals space 70 degree
straight i. straight e. space angle ACB space equals space 70 degree
Hence comma space correct space Option space is space left parenthesis straight c right parenthesis. end style

Question 7

In a triangle, an exterior angle at a vertex is 95° and its one of the interior opposite angle is 55°, then the measure of the other interior angle is

(a) 55°

(b) 85°

(c) 40°

(d) 9.0°

Solution 7

begin mathsize 12px style Let space the space other space interior space opposite space angle space be space straight x degree.
Then comma space we space have
straight x degree plus 55 degree equals 95 degree
rightwards double arrow straight x degree equals 95 degree minus 55 degree equals 40 degree
Hence comma space corrcet space option space is space left parenthesis straight c right parenthesis. end style

Question 8

If the sides of a triangle are produced in order, then the sum of the three exterior angles so formed is

(a) 90°

(b) 180°

(c) 270°

(d) 360°

Solution 8

begin mathsize 12px style 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
Now comma space angle FAB space equals space 180 degree space minus space angle straight A space space space.... open parentheses 1 close parentheses
space space space space space space space space space space space angle DCA space equals space 180 degree space minus space angle straight C space space.... open parentheses 2 close parentheses
space space space space space space space space space space space angle EBC space equals space 180 degree space minus space angle straight B space space space.... open parentheses 3 close parentheses
Adding space equations space open parentheses 1 close parentheses comma space open parentheses 2 close parentheses space and space open parentheses 3 close parentheses
angle FAB space plus space angle DCA space plus space angle EBC space equals space 180 degree space minus angle straight A space plus space 180 degree space minus space angle straight C space plus space 180 degree space minus space angle straight B
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 space space space space space space space space equals space 540 degree space minus space left parenthesis angle straight A space plus space angle straight B space plus space angle straight C right parenthesis
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 space space space space space space space space equals space 540 degree space minus 180 degree
rightwards double arrow Sum space of space All space exterior space angles space equals space 360 degree
Hence comma space correct space option space is space left parenthesis straight d right parenthesis. end style

Question 9

begin mathsize 12px style In space triangle ABC comma space if space angle straight A space equals space 100 degree comma space AD space bisects space angle straight A space and space AD space perpendicular space BC. space Then comma space angle straight B space equals
left parenthesis straight a right parenthesis space 50 degree
left parenthesis straight b right parenthesis space 90 degree
left parenthesis straight c right parenthesis space 40 degree
left parenthesis straight d right parenthesis space 100 degree end style

Solution 9

begin mathsize 12px style AD space perpendicular space BC space and space AD space bisects space angle straight A.
rightwards double arrow angle BAD space equals space angle CAD space equals space 50 degree
In space Right space triangle ADB
angle BAD space equals space 50 degree comma space space angle ADB space equals space 90 degree
Also space sum space of space all space interior space angles space equals space 180 degree
rightwards double arrow angle BAD space plus space angle space ADB space plus space angle straight B space equals space 180 degree
rightwards double arrow angle straight B space equals 180 degree space minus space 50 degree space minus space 90 degree
rightwards double arrow angle straight B space equals space 40 degree
Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 10

An exterior angle of a triangle is 108° and its interior opposite angles are in the ratio 4 : 5. The angles of the triangle are

(a) 48°, 60°, 72°

(b) 50°, 60°, 70°

(c) 52°, 56°, 72°

(d) 42°, 60°, 76°

Solution 10

begin mathsize 12px style From space figure comma space we space have
angle straight A space plus space angle straight B equals angle ACD space
rightwards double arrow 4 straight x degree space plus space 5 straight x degree equals space 108 degree
rightwards double arrow 9 straight x degree space equals space 108 degree
rightwards double arrow straight x space equals 12 degree
So comma space angle straight A space equals space 48 degree comma space angle straight B space equals space 60 degree space
rightwards double arrow angle straight C space equals space 180 degree space minus space 48 degree space minus space 60 degree space equals space 72 degree
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 11

begin mathsize 12px style In space straight a space triangle ABC comma space if space angle straight A space equals space 60 degree comma space angle straight B space equals space 80 degree space and space the space bisectors space of space angle straight B space and space angle straight C space meet space at space at space straight O comma space then space angle BOC space equals
left parenthesis straight a right parenthesis space 60 degree
left parenthesis straight b right parenthesis space 120 degree
left parenthesis straight c right parenthesis space 150 degree
left parenthesis straight d right parenthesis space 30 degree end style

Solution 11

begin mathsize 12px style straight O space is space point space where space bisectors space of space angle straight C space & space angle straight B space meets.
angle straight A space plus space angle straight B space plus space angle straight C space equals 180 degree
60 degree space plus space 80 degree space plus space angle straight C space equals space 180 degree
angle straight C space equals space 40 degree
fraction numerator angle straight C over denominator 2 end fraction equals 20 degree
fraction numerator angle straight C over denominator 2 end fraction equals 20 degree equals angle BCO space...... open parentheses 1 close parentheses
fraction numerator angle straight B over denominator 2 end fraction equals fraction numerator 80 degree over denominator 2 end fraction equals 40 degree equals angle OBC space...... open parentheses 2 close parentheses
In space triangle BOC
angle BCO space plus space angle OBC space plus space angle BOC space equals space 180 degree
From space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis space
20 degree space plus space 40 degree space plus space angle BOC space equals space 180 degree
rightwards double arrow angle BOC space equals space 180 degree space minus space 60 degree space equals space 120 degree
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 12

begin mathsize 12px style Line space sgements space AB space and space CD space intersect space at space straight O space such space that space AC space parallel to space DB. space If space angle CAB space equals space 45 degree space and
angle CDB space equals space 55 degree comma space then space angle BOD space equals
left parenthesis straight a right parenthesis space 100 degree
left parenthesis straight b right parenthesis space 80 degree
left parenthesis straight c right parenthesis space 90 degree
left parenthesis straight d right parenthesis space 135 degree end style

Solution 12

begin mathsize 12px style AC space parallel to space BD
And comma space AB space is space transverse space to space these space parallel space lines
So space angle CAB space equals space angle ABD space left parenthesis Alternate space angles right parenthesis
rightwards double arrow angle ABD space equals space 45 degree
Now space In space triangle BOD
angle BOD space plus space angle ODB space plus space angle DBA space equals space 180 degree
angle DBA space equals space angle ABD space equals space 45 degree comma space angle ODB space equals space 55 degree
So space angle BOD space equals space 180 degree space minus space 45 degree space minus space 55 degree
space space space space space space space space space space space space space space space space space space space space equals 80 degree
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Chapter 11 - Triangle and its Angles Excercise 11.26

Question 1

begin mathsize 12px style In space figure comma space if space EC space parallel to space AB comma space angle ECD space equals space 70 degree space and space angle BDO space equals space 20 degree comma space then space angle OBD space is
left parenthesis straight a right parenthesis space 20 degree
left parenthesis straight b right parenthesis space 50 degree
left parenthesis straight c right parenthesis space 60 degree
left parenthesis straight d right parenthesis space 70 degree end style

Solution 1

begin mathsize 12px style EC space parallel to space AB space and space CD space is space transverse space to space it.
Now space angle ECD space equals space angle AOD space equals space 70 degree space left parenthesis Corresponding space angles right parenthesis
In space triangle OBD
angle OBD space plus space angle BOD space plus space angle ODB space equals space 180 degree
angle BOD space equals space 180 degree space minus space angle AOD space equals space 180 degree space minus space 70 degree space equals 110 degree
angle ODB space equals space 20 degree space left parenthesis Given right parenthesis
So space angle OBD space equals space 180 degree space minus space angle BOD space minus space angle ODB
space space space space space space space space space space space space space space space space space space space equals space 180 degree space minus space 110 degree space minus space 20 degree
space space space space space space space space space space space space space space space space space space space equals space 50 degree
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 2

In figure, x + y =

(a) 270°

(b) 230°

(c) 210°

(d) 190°

Solution 2

begin mathsize 12px style In space triangle ACO
angle ACO space plus space angle COA space plus space angle OAC space equals space 180 degree
Now comma space angle OAC space equals space 180 space minus space straight x degree
rightwards double arrow 80 degree space plus space 40 degree space plus space up diagonal strike 180 space minus space straight x degree space equals space up diagonal strike 180 degree end strike
rightwards double arrow straight x degree space equals space 120 degree
angle BOD space equals space angle COA space equals space 40 degree space left parenthesis Opposite space angles right parenthesis
angle BDO space equals space 70 degree
In space triangle OBD comma
angle OBD space equals space 180 degree space minus space 40 degree space minus space 70 degree equals space 70 degree
Also comma space straight y degree space equals space 180 degree space minus space angle OBD space equals space 180 degree space minus space 70 degree equals 110 degree
rightwards double arrow straight x degree space plus space straight y degree space equals space 120 degree space plus space 110 degree space equals space 230 degree
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 3

If the measures of angles of a triangle are in the ratio of 3 : 4 : 5, what is the measure of the smallest angle of the triangle?

(a) 25°

(b) 30°

(c) 45°

(d) 60°

Solution 3

begin mathsize 12px style The space measures space of space angles space of space straight a space triangle space are space in space ratio space 3 space colon space 4 space colon space 5.
Let space the space angles space be space 3 straight x comma space 4 straight x space and space 5 straight x.
In space any space triangle comma space sum space of space all space angles space equals space 180 degree
rightwards double arrow 3 straight x space plus space 4 straight x space plus space 5 straight x space equals space 180 degree
rightwards double arrow 12 straight x equals space 180 degree
rightwards double arrow straight x space equals space 15 degree
So comma space smallest space angle space space equals space 3 space cross times space 15 degree space equals space 45 degree
Hence comma space correct space option space is space left parenthesis straight c right parenthesis.
end style

Question 4

begin mathsize 12px style In space figure comma space if space AB space perpendicular space BC comma space then space straight x space equals
left parenthesis straight a right parenthesis space 18
left parenthesis straight b right parenthesis space 22
left parenthesis straight c right parenthesis space 25
left parenthesis straight d right parenthesis space 32 end style

Solution 4

begin mathsize 12px style AB space perpendicular space BC
rightwards double arrow angle ABC space equals space 90 degree
angle CAB space equals space 32 degree space open parentheses Opposite space angles close parentheses
Now comma space in space triangle ABD
angle DAB space equals space straight x degree space plus space 32 degree
angle ABD space equals space 90 degree
angle BDA space equals space straight x degree space plus space 14 degree
In space straight a space triangle comma space sum space of space all space angles space equals space 180 degree
rightwards double arrow angle DAB space plus space angle ABD space plus space angle BDA space equals space 180 degree
rightwards double arrow straight x degree space plus space 32 degree space plus space 90 degree space plus space straight x degree space plus space 14 degree space equals space 180 degree
rightwards double arrow 2 straight x degree space equals space 180 degree space minus space 136 degree
rightwards double arrow 2 straight x degree space equals space 44
rightwards double arrow straight x degree space equals space 22 degree
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Chapter 11 - Triangle and its Angles Excercise 11.27

Question 1

In figure, what is z in terms of x and y?

(a) x +y + 180°

(b) x + y - 180°

(c) 180° - (x + y)

(d) x + y + 360°

Solution 1

begin mathsize 12px style From space figure
angle straight A space equals space straight z degree
angle ACB space equals space 180 space minus space straight x degree
angle ABC equals space 180 space minus space straight y degree
Now comma space in space triangle ABC
angle straight A space plus space angle straight B space plus space angle straight C space equals space 180 degree
rightwards double arrow straight z degree space plus space up diagonal strike 180 space minus space straight y degree space plus space 180 space minus space straight x degree space equals space up diagonal strike 180 degree end strike
rightwards double arrow straight z degree space equals space straight x degree space plus space straight y degree space minus space 180 degree
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 2

begin mathsize 12px style In space figure comma space for space which space value space of space straight x space is space straight l subscript 1 space parallel to space straight l subscript 2 ?
left parenthesis straight a right parenthesis space 37
left parenthesis straight b right parenthesis space 43
left parenthesis straight c right parenthesis space 45
left parenthesis straight d right parenthesis space 47 end style

 

Solution 2

begin mathsize 12px style Let space if space straight l subscript 1 space parallel to space straight l subscript 2 space and space AB space is space tranverse space to space it.
Then comma
angle PBA space should space be space equal space to space angle BAS space left parenthesis Alternate space angles right parenthesis
So space if space space straight l subscript 1 space parallel to space straight l subscript 2 comma space space then space angle BAS space equals 70 degree
rightwards double arrow angle BAC space equals space 78 degree space minus space 35 degree space equals space 43 degree space space space space.... open parentheses 1 close parentheses
Now comma space in space triangle ABC
straight x degree space plus space angle straight C space plus space angle BAC space equals space 180 degree
rightwards double arrow straight x degree space plus space 90 degree space plus space 43 degree space equals space 180 degree
rightwards double arrow straight x degree space equals space 180 degree space minus space 90 degree space minus space 43 degree space equals space 47 degree
rightwards double arrow straight x degree space equals space 47 degree
So space if space straight x degree space equals space 47 degree space then space space straight l subscript 1 space parallel to space straight l subscript 2 space
Hence comma space correct space option space is space left parenthesis straight d right parenthesis.
end style

Question 3

begin mathsize 12px style In space figure comma space what space is space straight y space in space terms space of space straight x ?
left parenthesis straight a right parenthesis space 3 over 2 straight x
left parenthesis straight b right parenthesis space 4 over 3 straight x
left parenthesis straight c right parenthesis space space straight x
left parenthesis straight d right parenthesis space 3 over 4 straight x end style

Solution 3

begin mathsize 12px style From space Figure comma
angle DOC space equals space 180 degree space minus space angle AOD space space left parenthesis Both space are space Supplementary right parenthesis
rightwards double arrow angle DOC space equals space 180 degree space minus space 3 straight y degree
Also comma space angle ACB space equals space 180 degree space minus space angle straight A space minus space angle straight B
rightwards double arrow angle ACB space equals space 180 degree space minus space straight x degree space minus space 2 straight x degree space equals space 180 degree space minus space 3 straight x degree
space And space angle ACD space equals space 180 degree space minus space angle ACB
space space space space space space space space space space space space space space space space space space space space space space space equals 180 degree space minus space left parenthesis 180 degree space minus space 3 straight x degree right parenthesis
rightwards double arrow angle ACD space equals space 3 straight x degree
Now comma space in space triangle OCD
angle DOC space plus space angle OCD space plus space angle straight D space equals 180 degree
180 degree space minus space 3 straight y degree space plus space 3 straight x degree space plus space straight y degree space equals space 180 degree space space space space left square bracket angle OCD equals angle ACD right square bracket
rightwards double arrow 2 straight y degree space equals space 3 straight x degree
rightwards double arrow straight y space equals space 3 over 2 straight x degree
space Hence comma space correct space option space is space left parenthesis straight a right parenthesis. space space space space space space space space space space space space space space space space space space space end style

Question 4

In figure, what is the value of x?

(a) 35

(b) 45

(c) 50

(d) 60

Solution 4

begin mathsize 12px style In triangle ABC comma
angle BCA space plus space angle CAB space plus space angle ABC space equals space 180 degree
rightwards double arrow 3 straight y degree space plus space straight x degree space plus space 5 straight y degree space equals space 180 degree
rightwards double arrow 8 straight y degree space plus space straight x degree space equals space 180 degree space space space space space... open parentheses 1 close parentheses space
straight A lso comma space 5 straight y degree space equals space 180 degree space minus space 7 straight y degree
rightwards double arrow 12 straight y degree space equals space 180 degree
rightwards double arrow straight y degree equals 15 degree
From space left parenthesis 1 right parenthesis comma space straight x degree space equals space 180 degree space minus space 8 straight y degree
rightwards double arrow straight x degree equals space 180 degree space minus space 8 space cross times space 15 degree
rightwards double arrow straight x degree space equals space 60 degree
Hence comma space correct space option space is space left parenthesis straight d right parenthesis. end style

Chapter 11 - Triangle and its Angles Excercise 11.28

Question 1

In figure, the value of x is

(a) 65°

(b) 80°

(c) 95°

(d) 120°

Solution 1

begin mathsize 12px style In space triangle ABD
angle straight A space plus space angle straight B space plus space angle straight D space equals space 180 degree
rightwards double arrow 55 degree space plus space angle DBA space plus space 25 degree space equals space 180 degree
rightwards double arrow angle DBA space equals space 180 degree space minus space 55 degree space minus space 25 degree
space space space space space space space space space space space space space space space space space space space equals 180 degree space minus space 80 degree
rightwards double arrow angle DBA space equals space 100 degree
So comma space angle DBC space equals space 180 degree space minus space angle DBA
space space space space space space space space space space space space space space space space space space space space space equals space 180 degree space minus space 100 degree
rightwards double arrow angle DBC space equals space 80 degree
Now comma space in space triangle EBC
angle straight E space plus space angle EBC space plus space angle straight C space equals space 180 degree
rightwards double arrow angle straight E space plus space 80 degree space plus space 40 degree space equals space 180 degree space space space space left parenthesis angle DBC equals angle EBC right parenthesis
rightwards double arrow angle straight E space equals space 180 degree space minus space 120 degree space equals space 60 degree
Also comma space straight x space equals space 180 degree space minus space angle straight E space equals 180 degree space minus space 60 degree
rightwards double arrow straight x space equals space 120 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 BP space parallel to space CQ space and space AC space equals space BC comma space then space the space measure space of space straight x space is
left parenthesis straight a right parenthesis space 20 degree
left parenthesis straight b right parenthesis space 25 degree
left parenthesis straight c right parenthesis space 30 degree
left parenthesis straight d right parenthesis space 35 degree end style

Solution 2

begin mathsize 12px style angle PBC space equals space angle QCD space space left parenthesis Corresponding space angles comma space OP space parallel to CQ space and space BC space is space transverse right parenthesis
rightwards double arrow angle PBC space equals space 70 degree
Now comma space angle PBA plus angle ABC equals angle PBC
rightwards double arrow 20 degree space plus space angle ABC space equals space 70 degree space space space space
rightwards double arrow angle ABC space equals space 50 degree
In space triangle ABC comma
angle ABC space plus space angle BAC space plus space angle ACB space equals space 180 degree space space space.... left parenthesis 1 right parenthesis
Now comma space angle ABC space equals space angle BAC space equals space 50 degree space left parenthesis isosceles space triangle right parenthesis space
And comma space angle ACB space equals space 180 degree space minus space left parenthesis 70 degree space plus space straight x right parenthesis
From space left parenthesis 1 right parenthesis comma
50 degree space plus space 50 degree space plus up diagonal strike space 180 degree end strike space minus space left parenthesis 70 degree space plus space straight x right parenthesis space equals space up diagonal strike 180 degree
rightwards double arrow straight x space equals space 30 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 figure comma space AB space and space CD space are space parallel to lines space and space transversal space EF space intersects space them space at space straight P space and space straight Q
respectively comma space If space angle APR space equals space 25 degree comma space angle RQC space equals space 30 degree space and space angle CQF space equals space 65 degree comma space then
left parenthesis straight a right parenthesis space straight x space equals space 55 degree comma space straight y space equals space 40 degree
left parenthesis straight b right parenthesis space straight x space equals space 50 degree comma space straight y space equals space 45 degree
left parenthesis straight c right parenthesis space straight x space equals space 60 degree comma space straight y space equals space 35 degree
left parenthesis straight d right parenthesis space straight x space equals space 35 degree comma space straight y space equals space 60 degree end style

Solution 3

begin mathsize 12px style angle OQP space equals space 180 degree space minus space angle OQF
space space space space space space space space space space space space space space equals space 180 degree space minus space left parenthesis 30 degree space plus space 65 degree right parenthesis
rightwards double arrow angle OQP space equals space 85 degree space space space space space.... open parentheses 1 close parentheses
angle APQ space equals space angle CQF space space space space left parenthesis Corresponding space angles right parenthesis
rightwards double arrow 25 degree space plus space straight y degree space equals space 65 degree
rightwards double arrow straight y degree space equals space 65 degree space minus space 25 degree
rightwards double arrow straight y degree space equals space 40 degree
Now space in space triangle OPQ
angle straight O space plus space angle OPQ space plus space angle PQO space equals space 180 degree
rightwards double arrow straight x degree space plus space 40 degree space plus space 85 degree space equals space 180 degree
straight x degree space equals space 180 space minus space 85 degree space minus space 40 degree space equals space 55 degree
rightwards double arrow space straight x degree space equals space 55 degree comma space straight y space equals space 40 degree
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 4

begin mathsize 12px style The space base space BC space of space triangle space ABC space is space produced space both space ways space and space the space measure space of space exterior space angles
formed space are space 94 degree space and space 126 degree. space Then comma space angle BAC space equals
left parenthesis straight a right parenthesis space 94 degree
left parenthesis straight b right parenthesis space 54 degree
left parenthesis straight c right parenthesis space 40 degree
left parenthesis straight d right parenthesis space 44 degree end style

Solution 4

begin mathsize 12px style angle ABC space equals space 180 degree space minus space 126 degree equals space 54 degree
angle ACB space equals space 180 space minus space 94 degree equals space 86 degree
Now comma space in space triangle ABC
angle BAC space plus space angle ABC space plus space angle ACB space equals space 180 degree
rightwards double arrow angle BAC space equals space 180 degree space minus space angle ABC space minus space angle ACB
space space space space space space space space space space space space space space space space space space space equals 180 degree space minus space 54 degree space minus space 86 degree
rightwards double arrow angle BAC space equals space 40 degree
Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Chapter 11 - Triangle and its Angles Excercise 11.29

Question 1

If the bisectors of the acute angles of a right triangle meet at O, then the angle at O between the two bisectors is

(a) 45°

(b) 95°

(c) 135°

(d) 90°

Solution 1

begin mathsize 12px style 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 90 degree space plus space angle straight C space equals space 180 degree
rightwards double arrow angle straight A space plus space angle straight C space equals space 90 degree space space space space.... open parentheses 1 close parentheses
Now comma space in space triangle AOC comma
angle COA space plus space angle OAC space plus space angle OCA space equals space 180 degree
rightwards double arrow angle COA space plus space fraction numerator angle straight A over denominator 2 end fraction plus space fraction numerator angle straight C over denominator 2 end fraction space equals space 180 degree space space open curly brackets AO space and space CO space bisects space angle space angle straight A space and space angle straight C close curly brackets
rightwards double arrow angle COA space equals 180 minus space open parentheses fraction numerator angle straight A plus angle straight C over denominator 2 end fraction close parentheses
space space space space space space space space space space space space space space space space space space space space equals space 180 degree space minus space open parentheses fraction numerator 90 degree over denominator 2 end fraction close parentheses space space space space space space open curly brackets From space open parentheses 1 close parentheses close curly brackets
space space space space space space space space space space space space space space space space space space space space equals space 100 degree space minus space 45 degree space
space space space space space space space space space space space space space space space space space space space space space equals space 135 degree
Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 2

begin mathsize 12px style The space bisects space of space exterior space angles space at space straight B space and space straight C space of space triangle ABC space meet space at space straight O. space If space angle straight A space equals space straight x degree comma space then space angle BOC equals
left parenthesis straight a right parenthesis space 90 degree plus fraction numerator straight x degree over denominator 2 end fraction
left parenthesis straight b right parenthesis space space 90 degree minus fraction numerator straight x degree over denominator 2 end fraction
left parenthesis straight c right parenthesis space space 180 degree plus fraction numerator straight x degree over denominator 2 end fraction
left parenthesis straight d right parenthesis space space 180 degree minus fraction numerator straight x degree over denominator 2 end fraction end style

Solution 2

begin mathsize 12px style 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 B space plus space angle straight C space equals space 180 space minus space straight x degree space space space space space.... open parentheses 1 close parentheses
angle CBD space equals space 180 degree space minus angle straight B space space space space.... open parentheses 2 close parentheses
angle ECB space equals space 180 degree space minus space angle straight C space space space.... open parentheses 3 close parentheses
rightwards double arrow fraction numerator angle CBD over denominator 2 end fraction equals angle OBC space equals 90 degree space minus fraction numerator angle straight B over denominator 2 end fraction space.... left parenthesis 4 right parenthesis space left square bracket From space eq space open parentheses 2 close parentheses right square bracket
space space space space space space fraction numerator angle ECB over denominator 2 end fraction equals angle OCB space equals 90 degree space minus fraction numerator angle straight C over denominator 2 end fraction space space....... left parenthesis 5 right parenthesis space space space space left square bracket From space eq space left parenthesis 3 right parenthesis right square bracket space
Now comma space in space triangle BOC
angle OBC space plus space angle OCB space plus space angle BOC space equals space 180 degree
rightwards double arrow angle BOC space equals space 180 degree space minus space left parenthesis angle OBC space plus space angle OCB right parenthesis
From space eq space open parentheses 4 close parentheses space and space open parentheses 5 close parentheses comma
angle BOC space equals space 180 degree space minus space open parentheses 90 degree space minus space fraction numerator angle straight B over denominator 2 end fraction space plus space 90 degree space minus space fraction numerator angle straight C over denominator 2 end fraction close parentheses
space space space space space space space space space space space space space space equals space 180 degree space minus space open parentheses 180 degree space minus fraction numerator angle straight B over denominator 2 end fraction minus fraction numerator angle straight C over denominator 2 end fraction close parentheses
space space space space space space space space space space space space space space equals fraction numerator angle straight B space plus space angle straight C over denominator 2 end fraction space space space space space space left square bracket From space eq space open parentheses 1 close parentheses right square bracket
space space space space space space space space space space space space space equals fraction numerator 180 degree space minus space straight x degree over denominator 2 end fraction
rightwards double arrow angle BOC space equals space 90 degree space minus fraction numerator straight x degree over denominator 2 end fraction
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 3

begin mathsize 12px style In space straight a space triangle ABC comma space angle straight A space equals space 50 degree space and space BC space is space produced space to space straight a space point space straight D. space If space the space bisectors space of space angle ABC space and space angle ACD space meet space at space straight E comma
then space angle straight E space equals
left parenthesis straight a right parenthesis space 25 degree
left parenthesis straight b right parenthesis space 50 degree
left parenthesis straight c right parenthesis space 100 degree
left parenthesis straight d right parenthesis space 75 degree end style

Solution 3

begin mathsize 12px style BE space and space CE space are space bisectors space of space angle straight B space and space angle ACD.
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 B space plus space angle straight C space equals space 180 degree space minus space 50 degree space equals space 130 degree space space space space.... open parentheses 1 close parentheses
Now comma space in space triangle BEC
angle CBE space plus space angle BEC space plus space angle ECB space equals space 180 degree space space space space... open parentheses 2 close parentheses
angle space CBE space equals space fraction numerator angle straight B over denominator 2 end fraction comma space space space angle BEC space equals space angle straight E comma space angle ECB space equals space angle straight C space plus space angle ACE
Now comma space angle ACD space equals space 180 degree space minus space angle straight C
angle ACE space equals space fraction numerator angle ACD over denominator 2 end fraction space equals space fraction numerator 180 degree space minus angle straight C over denominator 2 end fraction space equals space 90 degree space minus fraction numerator angle straight C over denominator 2 end fraction
So comma space angle ECB space equals space angle straight C space plus space space 90 degree space minus fraction numerator angle straight C over denominator 2 end fraction
rightwards double arrow angle ECB space equals space 90 degree space plus space fraction numerator angle straight C over denominator 2 end fraction
Now space putting space all space values space in space eq space open parentheses 2 close parentheses
fraction numerator angle straight B over denominator 2 end fraction plus angle straight E space plus space 90 degree space plus fraction numerator angle straight C over denominator 2 end fraction equals 180 degree
rightwards double arrow angle straight E space equals space 180 degree space minus space 90 degree space minus space open parentheses fraction numerator angle straight B space plus angle straight C over denominator 2 end fraction close parentheses
space space space space space space space space space space space space space equals 90 degree space minus space open parentheses fraction numerator angle straight B space plus angle straight C over denominator 2 end fraction close parentheses space
space space space space space space space space space space space space space equals space 90 degree space minus fraction numerator 130 degree over denominator 2 end fraction space space space space space left square bracket From space eq space open parentheses 1 close parentheses right square bracket
rightwards double arrow angle straight E space equals space 25 degree
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 4

begin mathsize 12px style The space side space BC space of space triangle ABC space is space produced space to space straight a space point space straight D. space The space bisector space of space angle straight A space meets space side space BC space in space straight L.
If space angle ABC space equals space 30 degree space and space angle ACD space equals space 115 degree comma space then space angle ALC space equals
left parenthesis straight a right parenthesis space 85 degree
left parenthesis straight b right parenthesis space 72 1 half degree
left parenthesis straight c right parenthesis space 145 degree
left parenthesis straight d right parenthesis space None end style

Solution 4

begin mathsize 12px style angle straight C space equals space 180 degree space minus space angle ACD equals space 180 degree space minus space 115 degree equals space 65 degree
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 equals space 180 space minus space 30 degree space minus space 65 degree
rightwards double arrow angle straight A space equals space 85 degree
Now space in space triangle ALC
angle ALC space plus space angle LAC space plus space angle straight C space equals space 180 degree
rightwards double arrow angle ALC space equals space 180 degree space minus space angle LAC space minus space angle straight C
space space space space space space space space space space space space space space space space space space equals space 180 degree space minus space fraction numerator angle straight A over denominator 2 end fraction minus angle straight C
space space space space space space space space space space space space space space space space space space equals 180 degree space minus space fraction numerator 85 degree over denominator 2 end fraction space minus space 65 degree
space space space space space space space space space space space space space space space space space space equals fraction numerator 145 degree over denominator 2 end fraction
space space space space space space space space space space space space space space space space space space equals 72 1 half degree
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 5

begin mathsize 12px style In space figure comma space if space straight l subscript 1 space parallel to space straight l subscript 2 comma space the space value space of space straight x space is
left parenthesis straight a right parenthesis space 22 1 half
left parenthesis straight b right parenthesis space 30
left parenthesis straight c right parenthesis space 45
left parenthesis straight d right parenthesis space 60 end style

Solution 5

begin mathsize 12px style From space figure comma
angle ACS space equals space 180 degree space minus space 2 straight b degree
Also space angle ACS space equals space angle PAC space equals space 2 straight a degree space space space space left parenthesis alternate space angles right parenthesis
rightwards double arrow 2 straight a degree space equals space 180 degree space minus space 2 straight b degree
rightwards double arrow straight a degree space plus space straight b degree space equals space 90 degree
Now comma space in space triangle ABC
straight a degree space plus space straight b degree space plus space angle ABC space equals space 180 degree
angle ABC space equals space 180 degree space minus space 2 straight x degree
rightwards double arrow straight a degree space plus space straight b degree space plus space up diagonal strike 180 degree end strike space minus space 2 straight x degree space equals space up diagonal strike 180 degree
rightwards double arrow 2 straight x degree space equals straight a degree space plus space straight b degree space equals space 90 degree
rightwards double arrow straight x degree space equals space 45 degree
Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 6

begin mathsize 12px style In space triangle RST comma space w hat space is space the space value space of space straight x ?
left parenthesis straight a right parenthesis space 40
left parenthesis straight b right parenthesis space 90 degree
left parenthesis straight c right parenthesis space 80 degree
left parenthesis straight d right parenthesis space 100 degree end style

Solution 6

begin mathsize 12px style In space triangle RST
angle straight R space plus space angle straight S space plus space angle straight T space equals space 180 degree
rightwards double arrow 2 straight a degree space plus space straight x degree space plus space 2 straight b degree space equals space 180 degree
rightwards double arrow straight x degree space equals space 180 degree space minus space 2 left parenthesis straight a space plus space straight b right parenthesis degree space space.... open parentheses 1 close parentheses
Now comma space in space triangle ROT
angle ORT space plus space angle ROT space plus space angle OTR space equals space 180 degree
rightwards double arrow straight a degree space plus space 140 degree space plus space straight b degree space equals space 180 degree
rightwards double arrow left parenthesis straight a space plus space straight b right parenthesis degree space equals space 180 degree space minus space 140 degree space equals space 40 degree space space space... open parentheses 2 close parentheses
From space eq space open parentheses 1 close parentheses space and space open parentheses 2 close parentheses
straight x degree space equals space 180 degree space minus space 2 left parenthesis 40 degree right parenthesis
rightwards double arrow straight x space equals space 100 degree
Hence comma space correct space option space is space left parenthesis straight d right parenthesis. end style

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