2.6 Sides opposite congruent angles
Theorem : If two sides of a triangle are equal, then the angles opposite them are also equal. This can be proven as follows :
Consider a D ABC where AB = AC ( figure 2.23 ).
Proof : To prove m Ð B = m Ð C drop a median from A to BC at point P. Since AP is the median, BP = CP.
\ In D ABP and D ACP
seg. AB @ seg. AC( given )
seg. BP @ seg. CP( P is midpoint )
seg. AP @ seg. AP( same line )
Therefore the two triangles are congruent by SSS postulate.
D ABP @ D ACP
\ m Ð B = m Ð C as they are corresponding angles of congruent triangles.
The converse of this theorem is also true and can be proven quite easily.
Consider D ABC where m Ð B = m Ð C ( figure 2.24 )
To prove AB = AC drop an angle bisector AP on to BC.
Since AP is a bisector m Ð BAP = m Ð CAP
m Ð ABP = m Ð ACP ( given )
seg. AP @ seg. AP (same side )
\ D ABP @ D ACP by AAS postulate.
Therefore the corresponding sides are equal.
\ seg. AB = seg. AC
Conclusion :If the two angles of a triangle are equal, then the sides opposite to them are also equal.