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)

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. GE 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.