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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 ).

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.


\ 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 )

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.




2.1 Introduction
2.2 Sum Of The Angles Of A Triangle
2.3 Types of Triangles
2.4 Altitude, Median And Angle Bisector
2.5 Congruence Of Triangles
2.6 Sides Opposite Congruent Angles

Chapter 3

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