Exercise 7.2

NCERT solutions for Class 9 Maths Triangles 

NCERT Solutions for Class 9 Mathematics Triangles

1. In an isosceles triangle ABC, with AB = AC, the bisectors of B and C intersect each other at O. Join A to O. Show that:

(i) OB = OC

(ii) AO bisects A.

Ans. (i) ABC is an isosceles triangle in which AB = AC.

C = B [Angles opposite to equal sides]

OCA + OCB = OBA + OBC

OB bisects B and OC bisects C

OBA = OBC and OCA = OCB

OCB + OCB = OBC + OBC

2OCB = 2OBC

OCB = OBC

Now in OBC,

OCB = OBC [Prove above]

OB = OC [Sides opposite to equal sides]

(ii) In AOB and AOC,

AB = AC [Given]

OBA = OCA [Given]

And B = C

B = C

OBA = OCA

OB = OC [Prove above]

AOB AOC [By SAS congruency]

OAB = OAC [By C.P.C.T.]

Hence AO bisects A.

NCERT Solutions for Class 9 Maths Exercise 7.2

2. In ABC, AD is the perpendicular bisector of BC (See figure). Show that ABC is an isosceles triangle in which AB = AC.

Ans. In AOB and AOC,

BD = CD [AD bisects BC]

ADB = ADC = [AD BC]

AD = AD [Common]

ABD ACD [By SAS congruency]

AB = AC [By C.P.C.T.]

Therefore, ABC is an isosceles triangle.

NCERT Solutions for Class 9 Maths Exercise 7.2

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

Ans. In ABE and ACF,

A= A [Common]

AEB = AFC = [Given]

AB = AC [Given]

ABE ACF [By ASA congruency]

BE = CF [By C.P.C.T.]

Altitudes are equal.

NCERT Solutions for Class 9 Maths Exercise 7.2

4. ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (See figure). Show that:

(i) ABE ACF

(ii) AB = AC or ABC is an isosceles triangle.

Ans. (i) In ABE and ACF,

A= A [Common]

AEB = AFC = [Given]

BE = CF [Given]

ABE ACF [By ASA congruency]

(ii) Since ABE ACF

BE = CF [By C.P.C.T.]

ABC is an isosceles triangle.

NCERT Solutions for Class 9 Maths Exercise 7.2

5. ABC and DBC are two isosceles triangles on the same base BC (See figure). Show that ABD = ACD.

Ans. In isosceles triangle ABC,

AB = AC [Given]

ACB = ABC …….(i) [Angles opposite to equal sides]

Also in Isosceles triangle BCD.

BD = DC

BCD = CBD ……….(ii) [Angles opposite to equal sides]

Adding eq. (i) and (ii),

ACB + BCD = ABC + CBD

ACD = ABD

Or ABD = ACD

NCERT Solutions for Class 9 Maths Exercise 7.2

6. ABC is an isosceles triangle in which AB = AC. Side BA is produced to D such that AD = AB. Show that BCD is a right angle (See figure).

Ans. In isosceles triangle ABC,

AB = AC [Given]

ACB = ABC …….(i) [Angles opposite to equal sides]

Now AD = AB [By construction]

But AB = AC [Given]

AD = AB = AC

AD = AC

Now in triangle ADC,

AD = AC

ADC = ACD ………(ii) [Angles opposite to equal sides]

Since BAC + CAD = ………(iii) [Linear pair]

And Exterior angle of a triangle is equal to the sum of its interior opposite angles.

In ABC,

CAD = ABC + ACB = ACB + ACB [Using (i)]

CAD = 2ACB ……….(iv)

Similarly, for ADC,

BAC = ACD + ADC

= ACD + ACD = 2 ACD ……….(v)

From eq. (iii), (iv) and (v),

2ACB + 2ACD =

2(ACB + ACD) =

ACB + ACD =

BCD =

Hence BCD is a right angle.

NCERT Solutions for Class 9 Maths Exercise 7.2

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

Ans. ABC is a right triangle in which,

A = And AB = AC

In ABC,

AB = AC

C = B ……….(i)

We know that, in ABC,

A + B + C = [Angle sum property]

B + B =

[A = (given) and B = C (from eq. (i)]

2B =

B =

Also C = [B = C]

NCERT Solutions for Class 9 Maths Exercise 7.2

8. Show that the angles of an equilateral triangle are each.

Ans. Let ABC be an equilateral triangle.

AB = BC = AC

AB = BC

C = A ……….(i)

Similarly, AB = AC

C = B ……….(ii)

From eq. (i) and (ii),

A = B = C ……….(iii)

Now in ABC

A + B + C = ……….(iv)

A + A + A =

3A =

A =

Since A = B = C [From eq. (iii)]

A = B = C =

Hence each angle of equilateral triangle is