
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 
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. 
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. 
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. 