ICSE Class 9 Maths Chapter 10 Isosceles Triangles

Chapter 10: Isosceles Triangles

10.1 Introduction

A triangle, with at least two sides equal to each other, is called an isosceles triangle.

If all the sides of a triangle are equal to each other, it is called an equilateral triangle.

An equilateral triangle satisfies all the properties of an isosceles triangle, whereas it is not necessary for an isosceles triangle to satisfy all the properties of an equilateral triangle.

Theorem 1

If two sides of a triangle are equal, the angles opposite to them are also equal.

Given: A triangle ABC in which AB = AC.

To Prove: angle B = angle C.

Construction: Draw AD perpendicular to BC.

Proof:

Hence Proved.

Theorem 2

If two angles of a triangle are equal, the sides opposite to them are also equal.

Given: A triangle ABC in which angle B = angle C.

To Prove: AB = AC.

Construction: Draw AD perpendicular to BC.

Proof:

Hence Proved.

Teacher's Note

When a door or window frame is damaged, carpenters check if it is still an isosceles triangle by measuring whether the two sides are equal. If they are equal, the angles at the base must be equal, ensuring the frame is balanced and level.

Properties to Prove

The bisector of the angle at the vertex of an isosceles triangle bisects the base at right angles.

If equal sides of an isosceles triangle are produced, the exterior angles so formed are equal.

The perpendicular bisector of the base of an isosceles triangle passes through the vertex of the triangle.

The line, joining the mid-point of the base of an isosceles triangle and the opposite vertex, is perpendicular to the base and bisects the angle at the vertex.

Worked Examples

Problem 1: Use the information given in the following figures to find the values of a, b and c:

Figure (i): AB is parallel to CD

Solution:

Since AB is parallel to CD and AD is transversal:

a = 36 degrees (Alternate angles)

In triangle CDE:

Exterior angle CEA = sum of two interior opposite angles = 32 degrees + 36 degrees = 68 degrees

Now, in triangle CEA, CE = CA

Therefore, b = angle CEA = 68 degrees

In triangle ACE, b + c + angle CEA = 180 degrees

68 degrees + c + 68 degrees = 180 degrees

c = 180 degrees - 136 degrees = 44 degrees

Figure (ii): AC bisects angle A

In triangle ABC, AB = AC, therefore angle ACB = angle ABC = 80 degrees

And, a + angle ACB + angle ABC = 180 degrees

a + 80 degrees + 80 degrees = 180 degrees

a = 20 degrees

Since AC bisects angle A: b = a = 20 degrees

In triangle ADC, AD = DC

Therefore, angle ACD = b = 20 degrees

And, b + c + angle ACD = 180 degrees

20 degrees + c + 20 degrees = 180 degrees

c = 140 degrees

Teacher's Note

Architects use angle bisectors in isosceles triangles when designing symmetric structures like roof trusses, where equal sides and angles ensure stability and proper weight distribution.

Problem 2: In the adjoining figure, AB = BC and AC = CD. Prove that: angle BAD : angle ADB = 3 : 1.

Solution:

Let angle ADB = x

In triangle ACD, AC = CD

Therefore, angle CAD = angle CDA = x and exterior angle ACB = angle CAD + angle CDA = x + x = 2x

In triangle ABC, AB = BC

Therefore, angle BAC = angle ACB = 2x

Therefore, angle BAD = angle BAC + angle CAD = 2x + x = 3x

And, angle BAD / angle ADB = 3x / x = 3 / 1

Therefore, angle BAD : angle ADB = 3 : 1

Hence Proved.

Teacher's Note

When constructing triangular braces for shelving or furniture, understanding angle ratios helps builders determine if the structure will be rigid and properly angled for weight support.

Problem 3: In the given figure, AD is perpendicular to BC and EF both. If angle EAB = angle FAC, show that triangles ABD and ACD are congruent. Also, find the values of x and y if AB = 2x + 3, AC = 3y + 1, BD = x and DC = y + 1.

Solution:

AD is perpendicular to EF

Therefore, angle EAD = angle FAD = 90 degrees

Given: angle EAB = angle FAC

Therefore, angle EAD - angle EAB = angle FAD - angle FAC

Therefore, angle BAD = angle CAD

In triangle ABD and triangle ACD, angle BAD = angle CAD

angle ADB = angle ADC = 90 degrees (Given AD is perpendicular to BC)

and AD = AD

Therefore, triangle ABD is congruent to triangle ACD (By ASA)

Hence Proved.

Triangle ABD is congruent to triangle ACD implies AB = AC and BD = CD (By C.P.C.T.C.)

Therefore, 2x + 3 = 3y + 1 and x = y + 1

Therefore, 2x - 3y = -2 and x - y = 1

On solving, we get: x = 5 and y = 4

Teacher's Note

Engineers use perpendicular bisectors when designing symmetrical support structures; if two sides and angles meet specific geometric conditions, they can determine exact measurements for material specifications.

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