Triangles are classified according to the number of sides and according to the angle measures. A Scalene triangle is one that has no two sides congruent. An Isosceles triangle is defined to be one which has at least two congruent sides. Due to this property, isosceles triangle possess some additional properties than the scalene triangles have.

Isosceles Triangle

In the above diagram, an isosceles triangle ABC is shown. The two congruent sides AB and AC are called legs of the isosceles triangle. The angle A formed by the intersection of the legs is called the vertex angle, the other two angles on the base B and C are called the base angles.

The isosceles triangle theorem is also known as Base angles theorem.Statement:
If two triangles are congruent, then the angles opposite to them are also congruent.

Proof of Isosceles theorem

Given: Δ PQR where
PQ $\cong$ PR
Prove: $\angle$ Q $\cong$ $\angle$ R
Isosceles Theorem
Statement Reasons
1. Let S be the mid point of QR 1. Every line segment has exactly one mid point.
2. Join P and S to get segment PS 2. Two points are sufficient to define a line.
3. QS $\cong$ RS 3. Definition of mid point.
4. PQ $\cong$ PR 4. Given
5. PS $\cong$ PS 5. Reflexive Property
6. Δ PQS $\cong$ Δ PRS 6. SSS criterion of Triangle Congruency
7. $\angle$Q $\cong$ $\angle$R 7. cpctc

The converse of isosceles theorem is also true.

Converse of Isosceles Theorem:
If two angles are congruent, then the sides opposite to them are also congruent.
The Perimeter and Area formulas used for any triangle is also true for an isosceles triangle.
Perimeter P = (a + b + c) units, where a, b and c represent the lengths of the triangle ABC.
In the case of an isosceles triangle two sides are of equal length, say b = c.

Hence the formula can be refined as: Perimeter P = a + 2b

The general area formula used for calculating the area of a triangle

Area = $\frac{1}{2}$ x base x height.

The Heron formula also used for calculating the area of a triangle
Δ = $\sqrt{s(s-a)(s-b)(s-c)}$
where s is the semi perimeter of the triangle and s = $\frac{a+b+c}{2}$
The formula can be simplified for an isosceles triangle where b = c as,
Δ = (s - b) $\sqrt{s(s - a)}$ and s can also be taken as s = $\frac{a}{2}$ + b.
In the case of as isosceles right triangle, using Pythagorean theorem, a relationship between the lengths of the legs and the hypotenuse can be written as:
c2 = a2 + b2 where c is the length of the hypotenuse and a and b are the lengths of the legs.

In case of an isosceles right triangle a = b = x (say)
Hence c2 = x2 + x2 = 2x2.
⇒ c = $\sqrt{2}$x.

In other word the length of hypotenuse in a 90-45-45 special right triangle is $\sqrt{2}$ times the length of the leg.

This result can be applied to the length of the diagonal of a square as, the length of the diagonal of a square is $\sqrt{2}$ times the side length.
Is an equilateral triangle isosceles?
By definition, an isosceles triangle should have at least two congruent sides. In an equilateral triangle all the three sides are congruent. Hence an equilateral triangle is a special isosceles triangle.

Solved Examples

Question 1: In the diagram given below BD $\cong$ AD $\cong$ AC  and m $\angle$ ABD = 32º

   Isosceles Triangle Problem

   Find the measures of a. $\angle$ BAD   b. $\angle$ ADB     c. $\angle$ ACD     d. $\angle$ CAD
Solution:
 
Triangles ABD and ACD can be identified as isosceles triangles.
   m $\angle$ BAD = m $\angle$ ABD = 32º                                 Base angles of isosceles triangle ABD.
   m $\angle$ ADB + m $\angle$BAD + m $\angle$ABD = 180º               Angles sum property of a triangle
   m $\angle$ ADB + 32o + 32o = 180º
   ⇒            m $\angle$ ADB = 116º
   m $\angle$ ADC = m $\angle$ ABD + m $\angle$BAD                        Exterior angle theorem.
                  = 32o + 32o = 64º
  m $\angle$ACD = m $\angle$ADC = 64º                                  Base angles of isosceles triangle ACD.
  m $\angle$ CAD + m $\angle$ ACD + m $\angle$ ADC = 180o              Angles sum property of a triangle
  m $\angle$ CAD + 64o + 64o = 180o
  m $\angle$ CAD = 52º
 

Question 2: In triangle ABC, AB $\cong$ AC. Find the values of x, y. Also calculate the lengths AB, AC and the measures of angles B and C.

  
Isosceles Triangle Example


Solution:
 
Given AB = AC
=> 2x - 1 = x + 5 
        or   x = 6    (Equation solved for x)
   => AB = AC = 2x - 1 = 2(6) - 1 = 11

The values of AB and AC is 11units.
Now, base angles of isosceles triangle ABC are equal.
  => 3y - 10 = y + 30            
        2y = 40
   ⇒    y = 20
   Hence m $\angle$B = m $\angle$ C = 3(20) - 10 = 60 - 10 = 50º.