Isosceles Triangle Theorem

Isosceles triangle is a classification done on the basis of the sides of a triangle. An isosceles triangle is defined to be a triangle with at least two congruent sides. This makes an equilateral triangle (all the three sides are congruent) also a special type of an isosceles triangle. Special names are given to the sides and the angles of an isosceles triangle.

Isosceles Triangle   
  • The congruent sides are called the legs

  • The non congruent side is called the base of the isosceles triangle.

  • The angles made on the base are called the base angles.

  • The angle made by the intersection of the congruent sides is called the vertical angle.

The Isosceles triangle theorem states the base angle property of an isosceles triangle.

                                                  Isosceles Triangle Theorem
 Isosceles Triangle Theorem Definition: If two sides of a triangle are congruent, then the angles opposite to those sides are congruent.
In the triangle ABC it is given the sides
AB $\cong$ AC

According to isosceles triangle theorem,
then
$\angle$ B $\cong$ $\angle$ C
       Isosceles Triangle Theorem

Isosceles Triangle Theorem Proof

A two column proof for the isosceles triangle theorem can be written as follows:

Isosceles Triangle Theorem Proof

Given: In  $\triangle $ABC AB $\cong$  AC

To Prove: $\angle$B $\cong$ $\angle$C.

               Statement 
                              Reason
1. Let D be the midpoint of BC
 1. Every line segment has exactly one midpoint.
 2. Construct the line segment AD.  2. Two points are sufficient to draw a line.
 3. AB $\cong$ AC  3. Given.
 4. AD $\cong$ AD  4. Reflexive Property
 5. BD $\cong$ CD  5. Definition of midpoint.
 6. $\triangle $ABD $\cong$ $\triangle $ ACD  6. SSS criteria of congruency.
 7. $\angle$B $\cong$ $\angle$C  7. CPCTC.

Let us solve an example problem applying the isosceles triangle theorem.

Isosceles Triangle Theorem Problem

In the above diagram, triangles PQR and TUR are isosceles and triangle SVR is equilateral.

Example:

Find the measures of angles marked as 1, 2, 3, 4 and 5.

m $\angle$ P = m $\angle$ Q = 48º.                      $\triangle $PQR is isosceles
m $\angle$ T = m $\angle$ U = 72º                       $\triangle $TUR is isosceles.   
m $\angle$ S = m $\angle$ V = m $\angle$ SRV = 60º        $\triangle $SVR is equilateral

It can be proved using triangle congruency,
$\angle$1 $\cong$ $\angle$5,   $\angle$2 $\cong$ $\angle$4.

m $\angle$ 3 = 180 -(72 + 72) = 36º
m $\angle$ 2 + m $\angle$ 3 + m $\angle$ 4 = m $\angle$ SRV = 60º
m $\angle$ 2 + m $\angle$ 4 = 60 - m $\angle$ 3 = 60 - 36 = 24
Hence m $\angle$ 2 = m $\angle$ 4 = 12º.

m $\angle$ 1 + m $\angle$ 2 + m $\angle$ 3 + m $\angle$ 4 + m $\angle$ 5 = m $\angle$PRQ = 180 - (48 + 48) = 84º
m $\angle$ 1 + m $\angle$ 5 = 84 - 60 = 24º
Hence m $\angle$ 1 = m$\angle$5 = 12º

Thus, m $\angle$1 = m $\angle$2 = m $\angle$4 = m $\angle$5 = 12º  and m $\angle$3 = 36º

Converse of the Isosceles Triangle Theorem

The converse of the isosceles triangle theorem is also true.

If two angles of a triangle are congruent, then the sides opposite to those angles are also congruent.

Converse of the Isosceles Triangle Theorem

The two column proof for the converse is as follows:

                     Statement
                                  Reason
 1. Draw AD $\perp$ to BC  1. Perpendicular can be drawn  to a line from a point
     which does not lie on the line.
 2. $\angle$B $\cong$  $\angle$C  2. Given
 3.  $\angle$BDA and $\angle$CDA are right angles.  3. Definition of a perpendicular.
 4.  $\angle$BDA $\cong$ $\angle$CDA  4. Right angles are congruent.
 5. AD $\cong$ AD  5. Reflexive Property
 6.  $\triangle $ABD $\cong$ $\triangle $ AC  6. AAS criteria of congruency.
 7. AB $\cong$ AC  7. CPCTC.

Example:

Find the perimeter of triangle ABC given below.
Perimeter of Triangle

In the diagram angles B and C are marked as congruent.

Hence triangle ABC is isosceles with B and C as base angles.

AB = AC = 8 cm  (By converse of Isosceles triangle theorem).

Perimeter of triangle ABC = AB + BC + CA = 8 + 4 + 8 = 20 cm.

Prove an equilateral triangle is also equiangular.
Equilateral Triangle

Given: Triangle ABC is equilateral.

To Prove: $\angle$A $\cong$ $\angle$B $\cong$ $\angle$C.

                Statement 
                                       Reason
 1. AB $\cong$ BC $\cong$ CA  1. All the three sides are congruent in an equilateral triangle
 2.  $\triangle $ABC is isosceles  2.  AB $\cong$ AC
 3. $\angle$B $\cong$ $\angle$C  3. By  Isosceles triangle theorem.
 4. $\angle$C $\cong$ $\angle$A  4.  . AB $\cong$ BC  (By isosceles triangle theorem)
 5. $\angle$B $\cong$ $\angle$A  5. Transitive property.
 6. $\angle$A $\cong$ $\angle$B $\cong$ $\angle$C  6. By statements 4 and 5.
 7.  $\triangle $ABC is equiangluar  7. Definition of equiangular triangle.

Both the isosceles triangle theorem and its converse are widely used in proving many other theorems and riders.