Congruency of Triangles

Introduction to Congruent Triangles

• If all the sides and angles of a triangle are equal to the corresponding sides and angles of another triangle, then both the triangles are said to be congruent.
  
  Here, $△ABC\cong △DEF$

Criteria for Congruency

SSS Criteria for Congruency

• If under a given correspondence, the three sides of one triangle are equal to the three corresponding sides of another triangle, then the triangles are congruent.
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SAS Criteria for Congruency

• If under a correspondence, two sides and the angle included between them of a triangle are equal to two corresponding sides and the angle included between them of another triangle, then the triangles are congruent.
  

ASA Criteria for Congruency

• If under a correspondence, two angles and the included side of a triangle are equal to two corresponding angles and the included side of another triangle, then the triangles are congruent.
  

AAS Criteria for Congruency

AAS Rule: Triangles are congruent if two pairs of corresponding angles and a pair of opposite sides are equal in both triangles.

Criteria for Congruency

Criteria for Congruency of two triangles are:
(i) SSS Rule
(ii) SAS Rule
(iii) ASA Rule
(iv) RHS Rule

 Question 1.

What additional information do you need to conclude that the two triangles given here under are congruent using SAS rule?

Solution:
We need
HG = TS
HJ = TR

Question 2.
The map given below shows five different villages. Village M lies exactly halfway between the two pairs of villages A and B as well as and P and Q. What is the distance between village A and village P. (Hint: check if ΔPAM ≅ ΔQBM)

Solution:
Given that
AM = MB
PM = MQ
also ∠PMA = ∠QMB
∴ By S.A.S
ΔPMA ≅ ΔQMB
Now PA = BQ
(∵ Corresponding parts of congruent triangles are equal)
∴ PA = 4 km
i.e., Distance between the villages
A and P is 4 km.

Question 3.
Look at the pairs of triangles given below. Are they congruent ? If congruent write the corresponding parts.