9th Maths 7.2

NCERT Class 9th solution of Exercise 7.1

NCERT Class 9th Maths Projects

Exercise 7.2

Q1. In an isosceles triangle

$ABC$ , with

$AB=AC$ , the bisectors of

$angleB$ and

$angleC$ intersect each other at

$O$ . Join

$A$ to

$O$ . Show that :

$OB=OC$ ii)

$AO$ bisects

$angleA$

Sol. :

i) In

$triangleABC$ m

$AB=AC$ [Given]

$angleABC=angleACB$ [Angles opposite to equal sides]

$angleOBC=angleOCB$ [

$OB$ and

$OC$ bisect

$angleB$ and

$angleC$ ]

$OB=OC$ [Sides opposite to equal angles]

Proved.

ii) In

$triangleAOB$ and

$triangleAOC$

$AB=AC$ [Given]

$OB=OC$ [Proved in (i)]

$OA=OA$ [Common]

$triangleAOB=triangleAOC$ [By

$SSS$ rule]

$angleOAB=angleOAC$ [CPCT]

So that,

$OA$ bisect

$angleA$ .

Proved.

Q2. In

$triangleABC$ ,

$AD$ is the perpendicular bisector of

$BC$ (see figure). Show that

$triangleABC$ is an isosceles triangle in which

$AB=AC$ .

Sol. :

$triangleADB$ and

$triangleADC$

$AD=AD$ [Common]

$angleADB=angleADC=90^circ$ [Given

$AD⊥BC$ ]

$BD=CD$ [Given

$AD$ bisect

$BC$ at

$D$ ]

$triangleADBcongtriangleADC$ [By

$SAS$ rule]

$AB=AC$ [CPCT]

Proved.

Q3.

$ABC$ is an isosceles triangle in which altitudes

$BE$ and

$CF$ are drawn to equal sides

$AC$ and

$AB$ respectively (see figure). Show that these altitudes are equal.

Sol. :

$triangleABE$ and

$triangleACF$

$angleBAE=angleCAF$ [Common]

$angleAEB=angleAFC=90^circ$ [Given

$BE⊥AC$ and

$CF⊥AB$ ]

$AB=AC$ [Given]

$triangleABEcongtriangleACF$ [By

$A$

$S$ rule]

$BE=CF$ [CPCT]

So that, altitudes are equal.

Proved.

Q4.

$ABC$ is a triangle in which altitudes

$BE$ and

$CF$ to sides

$AC$ and

$AB$ are equal (see figure). Show that

$triangleABEcongtriangleACF$

ii)

$AB=AC$ , i.e.

$ABC$ is an isosceles triangle.

Sol. :

$triangleABE$ and

$ACF$

$angleBAE=angleCAF$ [Common]

$angleBEA=angleCFA$ [

$BE⊥AC$ and

$CF⊥AB$ ]

$BE=CF$ [Given]

$triangleABEcongtriangleACF$ [By

$A$

$S$ rule]

ii)

$triangleABEcongtriangleACF$ [Proved in (i)]

$AB=AC$ [CPCT]

So that,

$ABC$ is an isosceles triangle.

Proved.

Q5.

$ABC$ and

$DBC$ are two isosceles triangles on the same base

$BC$ (see figure). Show that

$angleABC=angleACD$ .

Sol. :

In an isosceles

$triangleABC$

$AB=AC$ [Given]

$angleCBA=angleBCA$ _______(1)

[angles opposite to equal sides]

In an isosceles

$triangleDBC$

$DB=DC$ [Given]

$angleCBD=angleBCD$ _______(2)

[angles opposite to equal sides]

We add equation (1) and (2)

$angleCBA+angleCBD=angleBCA+angleBCD$

$angleABD=angleACD$

Proved.

Q6.

$triangleABC$ is an isosceles triangle in which

$AB=AC$ . Side

$BA$ is produced to

$D$ such that

$AD=AB$ (see figure). Show that

$angleBCD$ is a right angle.

Sol. :

$triangleABC$ ,

$AB=AC$

$angleABC=angleACB$ ________(1)

[Angles oppostie to equal sides]

$AB=AD$ and

$AB=AC$ [Given]

$AD=AC$

$angleADC=angleACD$ ________(2)

[Angles opposite to equal sides]

We add equation (1) and (2)

$angleABC+angleADC=angleACB+angleACD$

$angleDBC+angleBDC=angleBCD$ _____(3)

By angle sum property of

$triangle$

$angleDBC+angleBDC+angleBCD=180^circ$

$angleBCD+angleBCD=180^circ$ [from equation (3)]

$2angleBCD=180^circ$

$angleBDC=180^circ/2$

$angleBDC=90^circ$

So that

$angleBDC$ is a right angle.

Proved.

Q7.

$ABC$ is a right angled triangle in which

$angleA=90^circ$ and

$AB=AC$ . Find

$angleB$ and

$angleC$ .

Sol. :

$triangleABC$ ,

$AB=AC$ [Given]

$angleB=angleC$ _______(1)

[Angles opposite to equal sides]

By angle sum property of

$triangle$

$angleA+angleB+angleC=180^circ$

$90^circ+angleB+angleB=180^circ$

$2angleB=180^circ-90^circ$

$angleB=90^circ/2$

$angleB=45^circ$

So that,

Answer.

$angleB=angleC=45^circ$

Q8. Show that the angles of an equilateral triangle are

$60^circ$ each.

Sol. :

Let each angle of an equilateral triangle is

$x^circ$ .

By angle sum property of

$triangle$

$x^circ+x^circ+xcirc=180^circ$

$3x^circ=180^circ$

$x^circ=180^circ/3$

$x^circ=60^circ$

So that, angles of an equilateral triangle is equal to

$60^circ$ each.

Proved.

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9th Maths 7.2

NCERT Class 9th solution of Exercise 7.1

NCERT Class 9th Maths Projects

Exercise 7.2

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