Chapter 10.1 - Examples of Congruence of Triangles

In the previous section we saw the congruence of Lines and Angles. In this section we will discuss about Congruence of Triangles.

Congruence of Triangles

Fig.10.5(a) shows two triangles ABC and PQR. They have different orientations.

Fig.10.5

If we make a trace copy of PQR and place it over ABC in such a way that:
• Corner P is over Corner A 
• Corner Q is over Corner B
• Corner R is over Corner C
Then PQR will completely and exactly cover ABC. This is shown in fig.10.5(b). So ABC and PQR are congruent to each other.

Now consider placing the trace copy in such a way that Q is over A. This is shown in the fig.10.6(a) below:

Fig.10.6

We can see that now PQR does not cover ABC. The congruence is lost. Another case is shown in fig.(b). Here, Q is over C. In this case also, PQR does not cover ABC. The congruence is lost. We can make cardboard cut outs and try different options. We will find that, the congruence will be valid only when a particular set of correspondence is ensured:

• A must correspond to P. This is written as: A↔P
• B must correspond to Q. This is written as: B↔Q 
• C must correspond to R. This is written as: C↔R
The above can be written in one line as ABC↔PQR. When writing like this in one line, the order must be preserved. That is.,
• The first letter A in ABC should correspond to the first letter P in PQR
• The second letter B in ABC should correspond to the second letter Q in PQR
• The third letter C in ABC should correspond to the third letter R in PQR

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We will now see some solved examples based on the above discussion
Solved example 10.1
Given two angles ∠ABC and ∠XYZ, which are congruent to one another. If ∠ABC is 87^o, what is the measure of ∠XYZ?
Solution:
• The two angles are congruent to one another. So both of them have the same angle
• So the measure of ∠XYZ is also 87^o 
Solved example 10.2
If ΔABC ≅ ΔXYZ under the correspondence ABC↔YZX, write all the corresponding parts of the two triangles
Solution:
• Given correspondence is  ABC↔YZX
• Following the order in this correspondence, we can write the corresponding vertices:
A↔Y, B↔Z, C↔X
• From this, we can write the corresponding sides:
AB↔YZ, BC↔ZX, AC↔YX
• The congruence can be seen in the animation in fig.10.7 below. Corresponding sides are given the same colours:

Fig.10.7

Solved example 10.3
ΔXYZ≅ΔQRP. Then Write the corresponding parts of the following:
(i) ∠Y (ii) Segment XY (iii) ∠X (iv) segment QP
Solution:  
• We have: ΔXYZ≅ΔQRP
• So the correspondence of vertices are: X↔Q, Y↔R and Z↔P
• From this we can write the corresponding parts
(i) ∠Y corresponds to ∠R
(ii) Segment XY corresponds to QR
(iii) ∠X corresponds to ∠Q
(iv) Segment QP corresponds to XZ

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In the next section we will discuss about the criterion for the Congruence of Triangles.

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