Download Congruence of Triangles

Congruence of Triangles
Two triangles are said to be congruent, if all the corresponding parts are equal.
The symbol used for denoting congruence is  and  PQR  STU implies that
and also
i.e. corresponding angles and corresponding sides are equal.
In order to prove that two triangles are congruent, it is not always necessary to
show that all the six corresponding parts are equal. If certain basic requirements
are met the triangles are said to be congruent. These basic criterias are
embodied in the five postulates given below. These postulates are axiomatic and
are useful in proving the congruence of triangles.
S S S Postulate If all the sides of one triangle are congruent to the
corresponding sides of another triangle then the triangles are congruent.
seg. AB = seg. PQ , seg. BC = seg. QR and
seg. CA = seg. RP
 ABC  PQR by S S S.
S A S Postulate
If the two sides and the angle included in one triangle are congruent to the
corresponding two sides and the angle included in another triangle then the two
triangles are congruent.
seg. AB = seg. PQ , seg. BC = seg. QR and
m  ABC = m  PQR
 ABC  PQR by S A S postulate.
A S A Postulate
If two angles of one triangle and the side they include are congruent to the
corresponding angles and side of another triangle the two triangles are
congruent.
m  B + m  R m  L = m  P and seg. BC = seg. RP
 ABC  QRP by A S A postulate.
A A S Postulate
If two angles of a triangle and a side not included by them are congruent to the
corresponding angles and side of another triangle the two triangles are
congruent.
m A = m P m B = m Q and AC = PR
 ABC  PQR by A A S.
H S Postulate
This postulate is applicable only to right triangles. If the hypotenuse and any one
side of a right triangle are congruent to the hypotenuse and the corresponding
side of another right triangle then the two triangles are congruent.
Then hypotenuse AC = hypotenuse PR
Side AB = Side PQ
  ABC   PQR by HS postulate.
Example 1
Name the postulate to be used to prove the following pairs of triangles are
congruent.
a)
a)
b)
c)
c)
d)
e)
f)
g)
Solution :
a) S S S
b) S AS
c) A S A
d) A A S
e) H L
f) None as there is no AAA postulate.
g) None, there is no ASS postulate.
Example 2
Seg. MN  seg. MO , seg. PN  seg. PO, prove that  MNP MOP.
Solution :
Draw segment MP
In triangle MNP and MOP
seg. MN seg.MO
seg. PN  seg. PO
seg. MP  seg. MP
 MNP  MOP ...(S S S Postulate)
 MNP  MOP ....(as they are corresponding
angles of congruent triangles .)
Example 3
H and G are two points on congruent sides DE & DF of  DEF such that seg. DH
 seg. DG. Prove that seg. HF  seg. GE.
Solution :
In triangles DHF and DGE seg. DF  seg. DE
HDF  GDE ( same angle )
seg. DH  seg. DG
 DHF  DGE S A S postulate
 seg. HF  seg.  as they are corresponding sides of
congruent triangles.
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  ABC where AB = AC
Proof : To prove m  B = m  C drop a median from A to BC at point P. Since
AP is the median, BP = CP.
 In  ABP and  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.
 ABP  ACP
 m  B = m  C as they are corresponding angles
of congruent triangles.
The converse of this theorem is also true and can be proven quite easily.
Consider  ABC where m  B = m  C
To prove AB = AC drop an angle bisector AP on to BC.
Since AP is a bisector m  BAP = m  CAP
m  ABP = m  ACP ( given )
seg. AP  seg. AP (same side )
 ABP 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.