ISOSCELES EQUILATERAL AND SCALENE TRIANGLES

Isosceles Triangle :

A triangle is isosceles, if it has at least two congruent sides or two congruent angles.

Equilateral Triangle :

A triangle is equilateral, if all the three sides are congruent or all the three angles are congruent.

Scalene Triangle :

A triangle is scalene triangle, if it has three unequal sides.

Base Angles Theorem

If two sides of a triangle are congruent, then the angles opposite them are congruent.

If AB ≅ AC, then

∠B ≅ ∠C

Converse of the Base Angles Theorem

If two sides of a triangle are congruent, then the angles opposite them are congruent.

If B ≅ C, then

AB ≅ AC

Corollaries to the Above Theorems 

Corollary (i) :

If a triangle is equilateral, then it is equiangular.

Corollary (ii) :

If a triangle is equiangular, then it is equilateral.

The above two corollaries have been illustrated in the picture given below.

Solving Problems on Isosceles Equilateral and Scalene Triangles

Problem 1 :

Use the diagram of ΔABC shown below to prove the Base Angles Theorem.

Solution :

Given : In ΔABC, AB ≅ AC.

To prove : ∠B ≅ ∠C.

Proof :

(i)  Draw the bisector of ∠CAB.

(ii)  By construction, ∠CAD ≅ ∠BAD.

(iii)  We are given that AB ≅ AC. Also DA ≅ DA, by the Reflexive Property of Congruence.

(iii) Use the SAS Congruence Postulate to conclude that ΔADB ≅ ΔADC.

(iv)  Because corresponding parts of congruent triangles are congruent, it follows that ∠B ≅ ∠C.

Problem 2 :

In the diagram shown below,

(i) find the value of x

(ii) find the value of y

Solution (i) :

In the diagram shown above, x represents the measure of an angle of an equilateral triangle.

From the corollary given above, if a triangle is equilateral, then it is equiangular.

So, the measure of each angle in the equilateral triangle is x.

By the Triangle Sum Theorem, we have

x° + x° + x° = 180°

Simplify. 

3x = 180

Divide both sides by 3 to solve for x.

x = 60

Solution (ii) :

In the diagram shown above, the vertex angle forms a linear pair with a 60° angle, so its measure is 120°.

It has been illustrated in the diagram given below.

By the Triangle Sum Theorem, we have

120° + 35° + y° = 180°

Simplify.

155° + 2y° = 180°

155 + 2y = 180

Subtract 155 from both sides.

y = 25

Problem 3 :

In the diagram shown below,

(i) find the value of x

(ii) find the value of y

Solution (i) :

In the diagram shown above, x represents the measure of an angle of an equilateral triangle.

From the corollary given above, if a triangle is equilateral, then it is equiangular.

So, the measure of each angle in the equilateral triangle is x.

By the Triangle Sum Theorem, we have

x° + x° + x° = 180°

Simplify.

3x = 180

Divide both sides by 3 to solve for x.

x = 60

Solution (ii) :

In the diagram shown above, y represents the measure of a base angle of an isosceles triangle.

From the Base Angles Theorem, the other base angle has the same measure. The vertex angle forms a linear pair with a 60° angle, so its measure is 120°.

It has been illustrated in the diagram given below.

By the Triangle Sum Theorem, we have

120° + y° + y° = 180°

Simplify. 

120 + 2y = 180

Subtract 120 from both sides.

2y = 60

Divide both sides by 2.

y = 30

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