NCERT Solutions for Class 9 Maths Chapter 7 Triangles

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NCERT Solutions for Class 9 Maths Chapter 7 Triangles

Edited By Ramraj Saini | Updated on Sep 27, 2023 10:38 PM IST

NCERT Solutions for Class 9 Maths Chapter 7 Triangles

NCERT solutions for class 9 maths chapter 7 Triangles are provided here. These NCERT Solutions are developed by subject matter expert at careersers360 considering latest CBSE syllabus 2023. Also these provide step by step solutions to all NCERT problems in comprehensive and simple way therefore these are easy to understand and ultimately beneficial for exams. In Class 9 Maths NCERT Syllabus Triangles, you will learn something of a higher level. NCERT triangles class 9 questions and answers can be a good tool whenever you are stuck in any of the problems. This Class 9 NCERT book chapter will be covering the properties of triangles like congruence of triangles, isosceles triangle, etc in detail.

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NCERT Solutions for Class 9 Maths Chapter 7 Triangles
Triangles Class 9 Questions And Answers PDF Free Download
Triangles Class 9 Solutions - Important Formulae And Points
Triangles Class 9 NCERT Solutions (Intext Questions and Exercise)
NCERT Solutions For Class 9 - Chapter Wise
NCERT Solutions For Class 9 - Subject Wise
NCERT Solutions for Class 9 Maths
NCERT Books and NCERT Syllabus

NCERT Solutions for Class 9 Maths Chapter 7 Triangles

By using NCERT class 9 maths chapter 7 question answer, you can prepare 360 degree for your school as well as for the competitive examinations. Here you will get NCERT solutions for class 9 Maths also.

Triangles Class 9 Questions And Answers PDF Free Download

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Triangles Class 9 Solutions - Important Formulae And Points

Congruence:

Congruent refers to figures that are identical in all aspects, including their shapes and sizes. For example, two circles with the same radii or two squares with the same side lengths are considered congruent.

Congruent Triangles:

Two triangles are considered congruent if and only if one of them can be superimposed (placed or overlaid) over the other in such a way that they entirely cover each other.

>> Congruence Rules for Triangles:

Side-Angle-Side (SAS) Congruence:
Angle-Side-Angle (ASA) Congruence:
Angle-Angle-Side (AAS) Congruence:.
Side-Side-Side (SSS) Congruence:
Right-Angle Hypotenuse Side (RHS) Congruence:

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Free download NCERT Solutions for Class 9 Maths Chapter 7 Triangles for CBSE Exam.

Triangles Class 9 NCERT Solutions (Intext Questions and Exercise)

NCERT solutions for class 10 maths chapter 7 Triangles - Excercise: 7.1

Q1 In quadrilateral $\small ABCD$ , $\small AC=AD$ and $\small AB$ bisects $\small \angle A$ (see Fig.). Show that $\small \Delta ABC\cong \Delta ABD$ . What can you say about $\small BC$ and $\small BD$ ?

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Answer:

In the given triangles we are given that:-

(i) $\small AC=AD$

(ii) Further, it is given that AB bisects angle A. Thus $\angle$ BAC $=\ \angle$ BAD.

(iii) Side AB is common in both the triangles. $AB=AB$

Hence by SAS congruence, we can say that : $\small \Delta ABC\cong \Delta ABD$

By c.p.c.t. (corresponding parts of congruent triangles are equal) we can say that $BC\ =\ BD$

Q2 (i) $ABCD$ is a quadrilateral in which $\small AD=BC$ and $\small \angle DAB= \angle CBA$ (see Fig. ). Prove that

$\small \Delta ABD\cong \Delta BAC$ .

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Answer:

It is given that :-

(i) AD = BC

(ii) $\small \angle DAB= \angle CBA$

(iii) Side AB is common in both the triangles.

So, by SAS congruence, we can write :

$\small \Delta ABD\cong \Delta BAC$

Q2 (ii) $\small ABCD$ is a quadrilateral in which $\small AD=BC$ and $\small \angle DAB=\angle CBA$ (see Fig.). Prove that $\small BD=AC$

1640157066047

Answer:

In the previous part, we have proved that $\small \Delta ABD\cong \Delta BAC$ .

Thus by c.p.c.t. , we can write : $\small BD=AC$

Q2 (iii) $\small ABCD$ is a quadrilateral in which $\small AD=BC$ and $\small \angle DAB= \angle BAC$ (see Fig.). Prove that $\small \angle ABD= \angle BAC$ .

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Answer:

In the first part we have proved that $\small \Delta ABD\cong \Delta BAC$ .

Thus by c.p.c.t. , we can conclude :

$\small \angle ABD= \angle BAC$

Q3 $\small AD$ and $\small BC$ are equal perpendiculars to a line segment $\small AB$ (see Fig.). Show that $\small CD$ bisects $\small AB$ .

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Answer:

In the given figure consider $\Delta$ AOD and $\Delta$ BOC.

(i) AD = BC (given)

(ii) $\angle$ A = $\angle$ B (given that the line AB is perpendicular to AD and BC)

(iii) $\angle$ AOD = $\angle$ BOC (vertically opposite angles).

Thus by AAS Postulate, we have

$\Delta AOD\ \cong \ \Delta BOC$

Hence by c.p.c.t. we can write : $AO\ =\ OB$

And thus CD bisects AB.

Q4 $\small l$ and $\small m$ are two parallel lines intersected by another pair of parallel lines $\small p$ and $\small q$ (see Fig. ). Show that $\small \Delta ABC\cong \Delta CDA$ .

1640157162694

Answer:

In the given figure, consider $\Delta$ ABC and $\Delta$ CDA :

(i) $\angle\ BCA\ =\ \angle DAC$

(ii) $\angle\ BAC\ =\ \angle DCA$

(iii) Side AC is common in both the triangles.

Thus by ASA congruence, we have :

$\Delta ABC\ \cong \ \Delta CDA$

Q5 (i) Line $\small l$ is the bisector of an angle $\small \angle A$ and B is any point on $\small l$ . $\small BP$ and $\small BQ$ are perpendiculars from $\small B$ to the arms of $\small \angle A$ (see Fig.). Show that: $\small \Delta APB\cong \Delta AQB$

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Answer:

In the given figure consider $\small \Delta APB$ and ,

(i) $\angle P\ =\ \angle Q$ (Right angle)

(ii) $\angle BAP\ =\ \angle BAQ$ (Since it is given that I is bisector)

(iii) Side AB is common in both the triangle.

Thus AAS congruence, we can write :

$\small \Delta APB\cong \Delta AQB$

Q5 (ii) Line $\small l$ is the bisector of an angle $\small \angle A$ and $\small B$ is any point on $\small l$ . $\small BP$ and $\small BQ$ are perpendiculars from $\small B$ to the arms of $\small \angle A$ (see Fig. ). Show that: $\small BP=BQ$ or $\small B$ is equidistant from the arms of $\small \angle A$ .

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Answer:

In the previous part we have proved that .

Thus by c.p.c.t. we can write :

$BP\ =\ BQ$

Thus B is equidistant from arms of angle A.

Q6 In Fig, and $\small \angle BAD= \angle EAC$ . Show that $\small BC=DE$ .

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Answer:

From the given figure following result can be drawn:-

$\angle BAD\ =\ \angle EAC$

Adding $\angle DAC$ to the both sides, we get :

$\angle BAD\ +\ \angle DAC\ =\ \angle EAC\ +\ \angle DAC$

$\angle BAC\ =\ \angle EAD$

Now consider $\Delta ABC$ and $\Delta ADE$ , :-

(i) $AC\ =\ AE$ (Given)

(ii) $\angle BAC\ =\ \angle EAD$ (proved above)

(iii) $AB\ =\ AD$ (Given)

Thus by SAS congruence we can say that :

$\Delta ABC\ \cong \ \Delta ADE$

Hence by c.p.c.t., we can say that : $BC\ =\ DE$

Q7 (i) $\small AB$ is a line segment and $\small P$ is its mid-point. $\small D$ and $\small E$ are points on the same side of $\small AB$ such that $\small \angle BAD=\angle ABE$ and $\small \angle EPA=\angle DPB$ (see Fig). Show that $\small \Delta DAP\cong \Delta EBP$

1640157265740

Answer:

From the figure, it is clear that :
$\angle EPA\ =\ \angle DPB$

Adding $\angle DPE$ both sides, we get :

$\angle EPA\ +\ \angle DPE =\ \angle DPB\ +\ \angle DPE$

or $\angle DPA =\ \angle EPB$

Now, consider $\Delta DAP$ and $\Delta EBP$ :

(i) $\angle DPA =\ \angle EPB$ (Proved above)

(ii) $AP\ =\ BP$ (Since P is the midpoint of line AB)

(iii) $\small \angle BAD=\angle ABE$ (Given)

Hence by ASA congruence, we can say that :

$\small \Delta DAP\cong \Delta EBP$

Q7 (ii) AB is a line segment and P is its mid-point. D and E are points on the same side of AB such that $\small \angle BAD=\angle ABE$ and $\small \angle EPA=\angle DPA$ (see Fig). Show that $\small AD=BE$

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Answer:

In the previous part we have proved that $\small \Delta DAP\cong \Delta EBP$ .

Thus by c.p.c.t., we can say that :

$\small AD=BE$

Q8 (i) In right triangle ABC, right angled at C, M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D such that $\small DM=CM$ . Point D is joined to point B (see Fig.). Show that: $\small \Delta AMC\cong \Delta BMD$

1640157315976

Answer:

Consider $\Delta AMC$ and $\Delta BMD$ ,

(i) $AM\ =\ BM$ (Since M is the mid-point)

(ii) $\angle CMA\ =\ \angle DMB$ (Vertically opposite angles are equal)

(iii) $CM\ =\ DM$ (Given)

Thus by SAS congruency, we can conclude that :

$\small \Delta AMC\cong \Delta BMD$

Q8 (ii) In right triangle ABC, right angled at C, M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D such that $\small DM=CM$ . Point D is joined to point B (see Fig.). Show that: $\small \angle DBC$ is a right angle.

1640157364987

Answer:

In the previous part, we have proved that $\small \Delta AMC\cong \Delta BMD$ .

By c.p.c.t. we can say that : $\angle ACM\ =\ \angle BDM$

This implies side AC is parallel to BD.

Thus we can write : $\angle ACB\ +\ \angle DBC\ =\ 180^{\circ}$ (Co-interior angles)

and, $90^{\circ}\ +\ \angle DBC\ =\ 180^{\circ}$

or $\angle DBC\ =\ 90^{\circ}$

Hence $\small \angle DBC$ is a right angle.

Q8 (iii) In right triangle ABC, right angled at C, M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D such that $\small DM=CM$ . Point D is joined to point B (see Fig.). Show that: $\small \Delta DBC\cong \Delta ACB$

1640157384655

Answer:

Consider $\Delta DBC$ and $\Delta ACB$ ,

(i) $BC\ =\ BC$ (Common in both the triangles)

(ii) $\angle ACB\ =\ \angle DBC$ (Right angle)

(iii) $DB\ =\ AC$ (By c.p.c.t. from the part (a) of the question.)

Thus SAS congruence we can conclude that :

$\small \Delta DBC\cong \Delta ACB$

Q8 (iv) In right triangle ABC, right angled at C, M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D such that $\small DM=CM$ . Point D is joined to point B (see Fig.). Show that: $\small CM=\frac{1}{2}AB$

1640165513559

Answer:

In the previous part we have proved that $\Delta DBC\ \cong \ \Delta ACB$ .

Thus by c.p.c.t., we can write : $DC\ =\ AB$

$DM\ +\ CM\ =\ AM\ +\ BM$

or $CM\ +\ CM\ =\ AB$ (Since M is midpoint.)

or $\small CM=\frac{1}{2}AB$ .

Hence proved.

NCERT Class 9 Maths Chapter 7 Question Answer - Excercise: 7.2

Q1 (i) In an isosceles triangle ABC, with $\small AB=AC$ , the bisectors of $\small \angle B$ and $\small \angle C$ intersect each other at O. Join A to O. Show that : $\small OB=OC$

Answer:

In the triangle ABC,

Since AB = AC, thus $\angle B\ =\ \angle C$

or $\frac{1}{2}\angle B\ =\ \frac{1}{2}\angle C$

or $\angle OBC\ =\ \angle OCB$ (Angles bisectors are equal)

Thus $\small OB=OC$ as sides opposite to equal are angles are also equal.

Q1 (ii) In an isosceles triangle ABC, with $\small AB=AC$ , the bisectors of $\small \angle B$ and $\small \angle C$ intersect each other at O. Join A to O. Show that : AO bisects $\small \angle A$

Answer:

Consider $\Delta AOB$ and $\Delta AOC$ ,

(i) $AB\ =\ AC$ (Given)

(ii) $AO\ =\ AO$ (Common in both the triangles)

(iii) $OB\ =\ OC$ (Proved in previous part)

Thus by SSS congruence rule, we can conclude that :

$\Delta AOB\ \cong \ \Delta AOC$

Now, by c.p.c.t.,

$\angle BAO\ =\ \angle CAO$

Hence AO bisects $\angle A$ .

Q2 In $\Delta ABC$ , AD is the perpendicular bisector of BC (see Fig). Show that $\small \Delta ABC$ is an isosceles triangle in which $\small AB=AC$ .

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Answer:

Consider $\Delta$ ABD and $\Delta$ ADC,

(i) $AD\ =\ AD$ (Common in both the triangles)

(ii) $\angle ADB\ =\ \angle ADC$ (Right angle)

(iii) $BD\ =\ CD$ (Since AD is the bisector of BC)

Thus by SAS congruence axiom, we can state :

$\Delta ADB\ \cong \ \Delta ADC$

Hence by c.p.c.t., we can say that : $\small AB=AC$

Thus $\Delta ABC$ is an isosceles triangle with AB and AC as equal sides.

Q3 ABC is an isosceles triangle in which altitudes BE and CF are drawn to equal sides AC and AB respectively (see Fig.). Show that these altitudes are equal.

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Answer:

Consider $\Delta AEB$ and $\Delta AFC$ ,

(i) $\angle A$ is common in both the triangles.

(ii) $\angle AEB\ =\ \angle AFC$ (Right angles)

(iii) $AB\ =\ AC$ (Given)

Thus by AAS congruence axiom, we can conclude that :

$\Delta AEB\ \cong \Delta AFC$

Now, by c.p.c.t. we can say : $BE\ =\ CF$

Hence these altitudes are equal.

Q4 (i) ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (see Fig). Show that $\small \Delta ABE \cong \Delta ACF$

1640165562618

Answer:

Consider $\Delta ABE$ and $\Delta ACF$ ,

(i) $\angle A$ is common in both the triangles.

(ii) $\angle AEB\ =\ \angle AFC$ (Right angles)

(iii) $BE\ =\ CF$ (Given)

Thus by AAS congruence, we can say that :

$\small \Delta ABE \cong \Delta ACF$

Q4 (ii) ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (see Fig.). Show that $\small AB=AC$ , i.e., ABC is an isosceles triangle.

1640157444769

Answer:

From the prevoius part of the question we found out that : $\Delta ABE\ \cong \Delta ACF$

Now, by c.p.c.t. we can say that : $AB\ =\ AC$

Hence $\Delta \ ABC$ is an isosceles triangle.

Q5 ABC and DBC are two isosceles triangles on the same base BC (see Fig.). Show that $\small \angle ABD\ \cong \ \angle ACD$ .

1640165593723

Answer:

Consider $\Delta ABD$ and $\Delta ACD$ ,

(i) $AD\ =\ AD$ (Common in both the triangles)

(ii) $AB\ =\ AC$ (Sides of isosceles triangle)

(iii) $BD\ =\ CD$ (Sides of isosceles triangle)

Thus by SSS congruency, we can conclude that :

$\small \angle ABD\ \cong \ \angle ACD$

Q6 $\Delta ABC$ is an isosceles triangle in which $AB=AC$ . Side BA is produced to D such that $AD=AB$ (see Fig.). Show that $\angle BCD$ is a right angle.

1640157465047

Answer:

Consider $\Delta$ ABC,
It is given that AB = AC

So, $\angle ACB = \angle ABC$ (Since angles opposite to the equal sides are equal.)

Similarly in $\Delta$ ACD,

We have AD = AB
and $\angle ADC = \angle ACD$
So,

$\angle CAB + \angle ACB + \angle ABC = 180^{\circ}$

$\angle CAB\ +\ 2\angle ACB = 180^{\circ}$
or $\angle CAB\ = 180^{\circ}\ -\ 2\angle ACB$ ...........................(i)

And in $\Delta$ ADC,
$\angle CAD\ = 180^{\circ}\ -\ 2\angle ACD$ ..............................(ii)

Adding (i) and (ii), we get :
$\angle CAB\ +\ \angle CAD\ = 360^{\circ}\ -\ 2\angle ACD\ -\ 2\angle ACB$

or $180^{\circ}\ = 360^{\circ}\ -\ 2\angle ACD\ -\ 2\angle ACB$

and $\angle BCD\ =\ 90^{\circ}$

Q7 ABC is a right angled triangle in which $\small \angle A =90^{\circ}$ and $\small AB=AC$ . Find $\small \angle B$ and $\small \angle C$ .

Answer:

In the triangle ABC, sides AB and AC are equal.

We know that angles opposite to equal sides are also equal.

Thus, $\angle B\ =\ \angle C$

Also, the sum of the interior angles of a triangle is $180^{\circ}$ .

So, we have :

$\angle A\ +\ \angle B\ +\ \angle C\ =\ 180^{\circ}$

or $90^{\circ} +\ 2\angle B\ =\ 180^{\circ}$

or $\angle B\ =\ 45^{\circ}$

Hence $\angle B\ =\ \angle C\ =\ 45^{\circ}$

Q8 Show that the angles of an equilateral triangle are $\small 60^{\circ}$ each.

Answer:

Consider a triangle ABC which has all sides equal.

We know that angles opposite to equal sides are equal.

Thus we can write : $\angle A\ =\ \angle B\ =\ \angle C$

Also, the sum of the interior angles of a triangle is $180 ^{\circ}$ .

Hence, $\angle A\ +\ \angle B\ +\ \angle C\ =\ 180^{\circ}$

or $3\angle A\ =\ 180^{\circ}$

or $\angle A\ =\ 60^{\circ}$

So, all the angles of the equilateral triangle are equal ( $60^{\circ}$ ).

Class 9 Triangles NCERT Solutions - Excercise: 7.3

Q1 (i) $\small \Delta ABC$ and $\small \Delta DBC$ are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC (see Fig.). If AD is extended to intersect BC at P, show that $\small \Delta ABD\cong \Delta ACD$

1640157485851

Answer:

Consider $\Delta ABD$ and $\Delta ACD$ ,

(i) $AD\ =\ AD$ (Common)

(ii) $AB\ =\ AC$ (Isosceles triangle)

(iii) $BD\ =\ CD$ (Isosceles triangle)

Thus by SSS congruency we can conclude that :

$\small \Delta ABD\cong \Delta ACD$

Q1 (ii) Triangles ABC and Triangle DBC are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC (see Fig). If AD is extended to intersect BC at P, show that $\small \Delta ABP \cong \Delta ACP$

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Answer:

Consider $\Delta ABP$ and $\Delta ACP$ ,

(i) $AP$ is common side in both the triangles.

(ii) $\angle PAB\ =\ \angle PAC$ (This is obtained from the c.p.c.t. as proved in the previous part.)

(iii) $AB\ =\ AC$ (Isosceles triangles)

Thus by SAS axiom, we can conclude that :

$\small \Delta ABP \cong \Delta ACP$

Q1 (iii) $\small \Delta ABC$ and $\small \Delta DBC$ are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC (see Fig.). If AD is extended to intersect BC at P, show that AP bisects $\small \angle A$ as well as $\small \angle D$ .

1640157546671

Answer:

In the first part, we have proved that $\small \Delta ABD\cong \Delta ACD$ .

So, by c.p.c.t. $\angle PAB\ =\ \angle PAC$ .

Hence AP bisects $\angle A$ .

Now consider $\Delta BPD$ and $\Delta CPD$ ,

(i) $PD\ =\ PD$ (Common)

(ii) $BD\ =\ CD$ (Isosceles triangle)

(iii) $BP\ =\ CP$ (by c.p.c.t. from the part (b))

Thus by SSS congruency we have :

$\Delta BPD\ \cong \ \Delta CPD$

Hence by c.p.c.t. we have : $\angle BDP\ =\ \angle CDP$

or AP bisects $\angle D$ .

Q1 (iv) $\small \Delta ABC$ and $\small \Delta DBC$ are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC (see Fig.). If AD is extended to intersect BC at P, show that AP is the perpendicular bisector of BC.

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Answer:

In the previous part we have proved that $\Delta BPD\ \cong \ \Delta CPD$ .

Thus by c.p.c.t. we can say that : $\angle BPD\ =\ \angle CPD$

Also, $BP\ =\ CP$

SInce BC is a straight line, thus : $\angle BPD\ +\ \angle CPD\ =\ 180^{\circ}$

or $2\angle BPD\ =\ 180^{\circ}$

or $\angle BPD\ =\ 90^{\circ}$

Hence it is clear that AP is a perpendicular bisector of line BC.

Q2 (i) AD is an altitude of an isosceles triangle ABC in which $\small AB=AC$ . Show that AD bisects BC

Answer:

Consider $\Delta ABD$ and $\Delta ACD$ ,

(i) $AB\ =\ AC$ (Given)

(ii) $AD\ =\ AD$ (Common in both triangles)

(iii) $\angle ADB\ =\ \angle ADC\ =\ 90^{\circ}$

Thus by RHS axiom we can conclude that :

$\Delta ABD\ \cong \ \Delta ACD$

Hence by c.p.c.t. we can say that : $BD\ =\ CD$ or AD bisects BC.

Q2 (ii) AD is an altitude of an isosceles triangle ABC in which $\small AB=AC$ . Show that AD bisects $\small \angle A$ .

Answer:

In the previous part of the question we have proved that $\Delta ABD\ \cong \ \Delta ACD$

Thus by c.p.c.t., we can write :

$\angle BAD\ =\ \angle CAD$

Hence $AD$ bisects $\angle A$ .

Q3 Two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of $\small \Delta PQR$ (see Fig). Show that:

(i) $\small \Delta ABM \cong \Delta PQN$

(ii) $\small \Delta ABC \cong \Delta PQR$

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Answer:

(i) From the figure we can say that :

$BC\ =\ QR$

or $\frac{1}{2}BC\ =\ \frac{1}{2}QR$

or $BM\ =\ QN$

Now, consider $\Delta ABM$ and $\Delta PQN$ ,

(a) $AM\ =\ PN$ (Given)

(b) $AB\ =\ PQ$ (Given)

Thus by SSS congruence rule, we can conclude that :

$\small \Delta ABM \cong \Delta PQN$

(ii) Consider $\Delta ABC$ and $\Delta PQR$ :

(a) $AB\ =\ PQ$ (Given)

(b) $\angle ABC\ =\ \angle PQR$ (by c.p.c.t. from the above proof)

Thus by SAS congruence rule,

$\small \Delta ABC \cong \Delta PQR$

Q4 BE and CF are two equal altitudes of a triangle ABC. Using RHS congruence rule, prove that the triangle ABC is isosceles.

Answer:

Using the given conditions, consider $\Delta BEC$ and $\Delta CFB$ ,

(i) $\angle BEC\ =\ \angle CFB$ (Right angle)

(ii) $BC\ =\ BC$ (Common in both the triangles)

(iii) $BE\ =\ CF$ (Given that altitudes are of the same length. )

Thus by RHS axiom, we can say that : $\Delta BEC\ \cong \Delta CFB$

Hence by c.p.c.t., $\angle B\ =\ \angle C$

And thus $AB\ =\ AC$ (sides opposite to equal angles are also equal). Thus ABC is an isosceles triangle.

Q5 ABC is an isosceles triangle with $\small AB=AC$ . Draw $\small AP\perp BC$ to show that $\small \angle B=\angle C$ .

Answer:

Consider $\Delta ABP$ and $\Delta ACP$ ,

(i) $\angle APB\ =\ \angle APC\ =\ 90^{\circ}$ (Since it is given that AP is altitude.)

(ii) $AB\ =\ AC$ (Isosceles triangle)

(iii) $AP\ =\ AP$ (Common in both triangles)

Thus by RHS axiom we can conclude that :

$\Delta ABP\ \cong \Delta ACP$

Now, by c.p.c.t.we can say that :

$\angle B\ =\ \angle C$

Class 9 maths chapter 7 NCERT solutions - Excercise: 7.4

Q1 Show that in a right-angled triangle, the hypotenuse is the longest side.

Answer:

Consider a right-angled triangle ABC with right angle at A.

We know that the sum of the interior angles of a triangle is 180.

So, $\angle A\ +\ \angle B\ +\ \angle C\ =\ 180^{\circ}$

or $90^{\circ}\ +\ \angle B\ +\ \angle C\ =\ 180^{\circ}$

or $\angle B\ +\ \angle C\ =\ 90^{\circ}$

Hence $\angle B$ and $\angle C$ are less than $\angle A$ ( $90^{\circ}$ ).

Also, the side opposite to the largest angle is also the largest.

Hence the side BC is largest is the hypotenuse of the $\Delta ABC$ .

Hence it is proved that in a right-angled triangle, the hypotenuse is the longest side.

Q2 In Fig, sides AB and AC of $\small \Delta ABC$ are extended to points P and Q respectively. Also, $\small \angle PBC < \angle QCB$ . Show that $\small AC> AB$ .

1640157763824

Answer:

We are given that,

$\small \angle PBC < \angle QCB$ ......................(i)

Also, $\angle ABC\ +\ \angle PBC\ =\ 180^{\circ}$ (Linear pair of angles) .....................(ii)

and $\angle ACB\ +\ \angle QCB\ =\ 180^{\circ}$ (Linear pair of angles) .....................(iii)

From (i), (ii) and (iii) we can say that :

$\angle ABC\ > \ \angle ACB$

Thus $AC\ > AB$ ( Sides opposite to the larger angle is larger.)

Q3 In Fig., $\small \angle B <\angle A$ and $\small \angle C <\angle D$ . Show that $\small AD <BC$ .

1640157791848

Answer:

In this question, we will use the property that sides opposite to larger angle are larger.

We are given $\small \angle B <\angle A$ and $\small \angle C <\angle D$ .

Thus, $BO\ > AO$ ..............(i)

and $OC\ > OD$ ...............(ii)

Adding (i) and (ii), we get :

$AO\ +\ OD\ <\ BO\ +\ OC$

or $AD\ <\ BC$

Hence proved.

Q4 AB and CD are respectively the smallest and longest sides of a quadrilateral ABCD (see Fig.). Show that $\small \angle A>\angle C$ and $\small \angle B>\angle D$ .

1640157840642

Answer:

1640157864575

Consider $\Delta ADC$ in the above figure :

$AD\ <\ CD$ (Given)

Thus $\angle CAD\ > \angle ACD$ (as angle opposite to smaller side is smaller)

Now consider $\Delta ABC$ ,

We have : $BC\ > AB$

and $\angle BAC\ > \angle ACB$

Adding the above result we get,

$\angle BAC\ +\ \angle CAD > \angle ACB\ +\ \angle ACD$

or $\small \angle A>\angle C$

Similarly, consider $\Delta ABD$ ,

we have $AB\ <\ AD$

Therefore $\angle ABD\ > \angle ADB$

and in $\Delta BDC$ we have,

$CD\ >\ BC$

and $\angle CBD\ >\ \angle CDB$

from the above result we have,

$\angle ABD\ +\ \angle CBD\ >\ \angle ADB\ +\ \angle CDB$

or $\small \angle B>\angle D$

Hence proved.

Q5 In Fig , $\small PR>PQ$ and PS bisects $\small \angle QPR$ . Prove that $\small \angle PSR>\angle PSQ$ .

1640165632055

Answer:

We are given that $\small PR>PQ$ .

Thus $\angle PQR\ =\ \angle PRQ$

Also, PS bisects $\small \angle QPR$ , thus :

$\angle QPS\ =\ \angle RPS$

Now, consider $\Delta QPS$ ,

$\angle PSR\ =\ \angle PQR\ +\ \angle QPS$ (Exterior angle)

Now, consider $\Delta PSR$ ,

$\angle PSQ\ =\ \angle PRQ\ +\ \angle RPS$

Thus from the above the result we can conclude that :

$\small \angle PSR>\angle PSQ$

Q6 Show that of all line segments drawn from a given point not on it, the perpendicular line segment is the shortest.

Answer:

Consider a right-angled triangle ABC with right angle at B.

Then $\angle B\ >\ \angle A\ or\ \angle C$ (Since $\angle B\ =\ 90^{\circ}$ )

Thus the side opposite to largest angle is also largest. $AC\ >\ BC\ or\ AB$

Hence the given statement is proved that all line segments are drawn from a given point, not on it, the perpendicular line segment is the shortest.

Triangles class 9 NCERT solutions - Excercise: 7.5

Q1 ABC is a triangle. Locate a point in the interior of $\small \Delta ABC$ which is equidistant from all the vertices of $\small \Delta ABC$ .

Answer:

We know that circumcenter of a triangle is equidistant from all the vertices. Also, circumcenter is the point of intersection of the perpendicular bisectors of the sides of a triangle.

Thus, draw perpendicular bisectors of each side of the triangle ABC. And let them meet at a point, say O.

Hence O is the required point which is equidistant from all the vertices.

Q2 In a triangle locate a point in its interior which is equidistant from all the sides of the triangle.

Answer:

The required point is called in-centre of the triangle. This point is the intersection of the angle bisectors of the interior angles of a triangle.

Hence the point can be found out in this case just by drawing angle bisectors of all the angles of the triangle.

Q3 In a huge park, people are concentrated at three points (see Fig.):

A : where there are different slides and swings for children,

B : near which a man-made lake is situated,

C : which is near to a large parking and exit.

Where should an icecream parlour be set up so that maximum number of persons can approach it? ( Hint : The parlour should be equidistant from A, B and C)

1640157898864

Answer:

The three main points form a triangle ABC. Now we have to find a point which is equidistant from all the three points.

Thus we need to find the circumcenter of the $\Delta ABC$ .

We know that circumcenter is defined as the point as the intersection point of the perpendicular bisectors of the sides of the triangle.

Hence the required point can be found out by drawing perpendicular bisectors of $\Delta ABC$ .

Q4 Complete the hexagonal and star shaped Rangolies [see Fig. (i) and (ii)] by filling them with as many equilateral triangles of side $\small 1$ cm as you can. Count the number of triangles in each case. Which has more triangles?

1640157916528

Answer:

For finding the number of triangles we need to find the area of the figure.

Consider the hexagonal structure :

Area of hexagon = 6 $\times$ Area of 1 equilateral

Thus area of the equilateral triangle :

$=\ \frac{\sqrt{3}}{4}\times a^2$

or $=\ \frac{\sqrt{3}}{4}\times 5^2$

or $=\ \frac{25\sqrt{3}}{4}\ cm^2$

So, the area of the hexagon is :

$=\6\times \frac{25\sqrt{3}}{4}\ =\ \frac{75\sqrt{3}}{2}\ cm^2$

And the area of an equilateral triangle having 1cm as its side is :

$=\ \frac{\sqrt{3}}{4}\times 1^2$

or $=\ \frac{\sqrt{3}}{4}\ cm^2$

Hence a number of equilateral triangles that can be filled in hexagon are :

$=\ \frac{\frac{75\sqrt{3}}{2}}{\frac{\sqrt{3}}{4}}\ =\ 150$

Similarly for star-shaped rangoli :

Area :

$=\12\times \frac{\sqrt{3}}{4}\times 5^2 \ =\ 75\sqrt{3}\ cm^2$

Thus the number of equilateral triangles are :

$=\ \frac{75\sqrt{3}}{\frac{\sqrt{3}}{4}}\ =\ 300$

Hence star-shaped rangoli has more equilateral triangles.

NCERT Triangles Class 9 Chapter 7 - Topics

Congruence of Triangles
Criteria for Congruence of Triangles
Some Properties of a Triangle
Some More Criteria for Congruence of Triangles
Inequalities in a Triangle

NCERT Solutions For Class 9 - Chapter Wise

Chapter No.	Chapter Name
Chapter 1	Number Systems
Chapter 2	Polynomials
Chapter 3	Coordinate Geometry
Chapter 4	Linear Equations In Two Variables
Chapter 5	Introduction to Euclid's Geometry
Chapter 6	Lines And Angles
Chapter 7	Triangles
Chapter 8	Quadrilaterals
Chapter 9	Areas of Parallelograms and Triangles
Chapter 10	Circles
Chapter 11	Constructions
Chapter 12	Heron’s Formula
Chapter 13	Surface Area and Volumes
Chapter 14	Statistics
Chapter 15	Probability

NCERT Solutions For Class 9 - Subject Wise

How To Use NCERT Solutions For Class 9 Maths Chapter 7 Triangles

Learn some basic properties of triangles using some books of previous classes.
Go through all the theorems and examples given in the chapter.
Once you have memorized all the theorems and properties, then you can practice the questions from practice exercises
During the practice, you can use NCERT solutions for class 9 maths chapter 7 triangles as a helpmate.
After doing all the exercises you can do some questions from past papers.

NCERT Books and NCERT Syllabus

Keep Working Hard and Happy Learning!

Frequently Asked Question (FAQs)

1. What are the important topics in chapter 7 maths class 9 ?

Congruence of triangles, Criteria for congruence of triangles, Properties of triangles, Inequalities of triangles are the important topics covered in this chapter. Students can practice NCERT solutions for class 9 maths to get command in these concepts that ultimately help during the exams.

2. In NCERT Solutions for class 9 chapter 7 maths, what does the concept of "congruence of triangles" signify?

NCERT Solutions for maths chapter 7 class 9 explain that "congruence of triangles" refers to the condition where two triangles are identical copies of each other and overlap perfectly when superimposed. In simpler terms, two triangles are considered congruent when the angles and sides of one triangle are equivalent to the corresponding angles and sides of the other triangle.

3. Where can I find the complete triangles class 9 solutions ?

Here you will get the detailed NCERT solutions for class 9 maths by clicking on the link. you can practice these solutions to command the concepts.

4. In what ways can NCERT Solutions for Class 9 Maths Chapter 7 assist in preparing for CBSE exams?

NCERT Solutions for Class 9 Maths Chapter 7 can assist students in achieving a high score and excelling in the subject in their CBSE exams. These solutions are designed based on the latest CBSE syllabus and cover all the essential topics in the respective subject. By practicing these solutions, students can gain confidence and be better prepared to face the board exams. The topics covered in these solutions are fundamental and contribute significantly to obtaining top scores. Moreover, solving problems of varying difficulty levels helps students get accustomed to answering questions of all types. Thus, these solutions are highly recommended for students as a reference and for practice in preparation for their CBSE exams.

5. How many chapters are there in the CBSE class 9 maths ?

There are 15 chapters starting from the number system to probability in the CBSE class 9 maths.

Also Read

NCERT Solutions for Exercise 2.5 Class 9 Maths Chapter 2 Polynomials

NCERT Solutions for Class 9 Maths Chapter 1 Exercise 1.1 Number Systems

NCERT Solutions for Exercise 14.3 Class 9 Maths Chapter 14 - Statistics

NCERT Solutions for Exercise 13.8 Class 9 Maths Chapter 13 - Surface Area and Volumes

NCERT Solutions for Exercise 13.7 Class 9 Maths Chapter 13 - Surface Area and Volumes

NCERT Solutions for Exercise 13.6 Class 9 Maths Chapter 13 - Surface Area and Volumes

NCERT Books for Class 9 2023 for All Subjects (Maths, Science, Social Science, Hindi English)

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NCERT Exemplar Class 9 Science Solutions Chapter 14 Natural Resources

NCERT Exemplar Class 9 Science Solutions Chapter 13 Why do we fall ill?

NCERT Exemplar Class 9 Science Solutions Chapter 12 Sound

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An SEO Analyst is a web professional who is proficient in the implementation of SEO strategies to target more keywords to improve the reach of the content on search engines. He or she provides support to acquire the goals and success of the client’s campaigns.

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Product Manager

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Quality Controller

A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.

A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.

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Production Manager

Production Manager Job Description: A Production Manager is responsible for ensuring smooth running of manufacturing processes in an efficient manner. He or she plans and organises production schedules. The role of Production Manager involves estimation, negotiation on budget and timescales with the clients and managers.

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Team Lead

A Team Leader is a professional responsible for guiding, monitoring and leading the entire group. He or she is responsible for motivating team members by providing a pleasant work environment to them and inspiring positive communication. A Team Leader contributes to the achievement of the organisation’s goals. He or she improves the confidence, product knowledge and communication skills of the team members and empowers them.

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Quality Systems Manager

A Quality Systems Manager is a professional responsible for developing strategies, processes, policies, standards and systems concerning the company as well as operations of its supply chain. It includes auditing to ensure compliance. It could also be carried out by a third party.

2 Jobs Available

Merchandiser

A career as a merchandiser requires one to promote specific products and services of one or different brands, to increase the in-house sales of the store. Merchandising job focuses on enticing the customers to enter the store and hence increasing their chances of buying a product. Although the buyer is the one who selects the lines, it all depends on the merchandiser on how much money a buyer will spend, how many lines will be purchased, and what will be the quantity of those lines. In a career as merchandiser, one is required to closely work with the display staff in order to decide in what way a product would be displayed so that sales can be maximised. In small brands or local retail stores, a merchandiser is responsible for both merchandising and buying.

2 Jobs Available

Procurement Manager

The procurement Manager is also known as Purchasing Manager. The role of the Procurement Manager is to source products and services for a company. A Procurement Manager is involved in developing a purchasing strategy, including the company's budget and the supplies as well as the vendors who can provide goods and services to the company. His or her ultimate goal is to bring the right products or services at the right time with cost-effectiveness.

2 Jobs Available

Production Planner

Individuals who opt for a career as a production planner are professionals who are responsible for ensuring goods manufactured by the employing company are cost-effective and meets quality specifications including ensuring the availability of ready to distribute stock in a timely fashion manner.

2 Jobs Available

ITSM Manager

ITSM Manager is a professional responsible for heading the ITSM (Information Technology Service Management) or (Information Technology Infrastructure Library) processes. He or she ensures that operation management provides appropriate resource levels for problem resolutions. The ITSM Manager oversees the level of prioritisation for the problems, critical incidents, planned as well as proactive tasks.

3 Jobs Available

Information Security Manager

Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack

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Computer Programmer

Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.

3 Jobs Available

Product Manager

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IT Consultant

An IT Consultant is a professional who is also known as a technology consultant. He or she is required to provide consultation to industrial and commercial clients to resolve business and IT problems and acquire optimum growth. An IT consultant can find work by signing up with an IT consultancy firm, or he or she can work on their own as independent contractors and select the projects he or she wants to work on.

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Data Architect

A Data Architect role involves formulating the organisational data strategy. It involves data quality, flow of data within the organisation and security of data. The vision of Data Architect provides support to convert business requirements into technical requirements.

2 Jobs Available

Security Engineer

The Security Engineer is responsible for overseeing and controlling the various aspects of a company's computer security. Individuals in the security engineer jobs include developing and implementing policies and procedures to protect the data and information of the organisation. In this article, we will discuss the security engineer job description, security engineer skills, security engineer course, and security engineer career path.

2 Jobs Available

UX Architect

A UX Architect is someone who influences the design processes and its outcomes. He or she possesses a solid understanding of user research, information architecture, interaction design and content strategy.

2 Jobs Available

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Option 1) $0.34\; J$	Option 2) $0.16\; J$
Option 3) $1.00\; J$	Option 4) $0.67\; J$

Option 1) $2,000 \; J - 5,000\; J$	Option 2) $200 \, \, J - 500 \, \, J$
Option 3) $2\times 10^{5}J-3\times 10^{5}J$	Option 4) $20,000 \, \, J - 50,000 \, \, J$

Option 1) $K/2\,$	Option 2) $\; K\;$
Option 3) $zero\;$	Option 4) $K/4$

Option 1) $11.2\, L\, H_{2(g)}$ at STP is produced for every mole $HCL_{(aq)}$ consumed	Option 2) $6L\, HCl_{(aq)}$ is consumed for ever $3L\, H_{2(g)}$ produced
Option 3) $33.6 L\, H_{2(g)}$ is produced regardless of temperature and pressure for every mole Al that reacts	Option 4) $67.2\, L\, H_{2(g)}$ at STP is produced for every mole Al that reacts .

Option 1) 0.02	Option 2) 3.125 × 10^-2
Option 3) 1.25 × 10^-2	Option 4) 2.5 × 10^-2

Option 1) decrease twice	Option 2) increase two fold
Option 3) remain unchanged	Option 4) be a function of the molecular mass of the substance.

Option 1) Molality	Option 2) Weight fraction of solute
Option 3) Fraction of solute present in water	Option 4) Mole fraction.

Option 1) twice that in 60 g carbon	Option 2) 6.023 × 10²²
Option 3) half that in 8 g He	Option 4) 558.5 × 6.023 × 10²³

Option 1) less than 3	Option 2) more than 3 but less than 6
Option 3) more than 6 but less than 9	Option 4) more than 9

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NCERT Solutions for Class 9 Maths Chapter 7 Triangles

NCERT Solutions for Class 9 Maths Chapter 7 Triangles

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