In the previous section we completed the discussion about the Pythagoras theorem. In this section we will discuss about Congruence of Triangles.
Fig.10.1(a) shows two line segments AB and PQ. These two segments are drawn in two different directions. So it is not easy to compare them.
If we make a trace copy of PQ and place it over AB, then PQ will completely cover AB. Not just 'complete cover', but it will be an 'exact cover' (as shown in fig.10.1.b) . That means, PQ is an exact copy of AB. It is neither more nor less than AB. It has the same shape and size as AB. So it will be able to cover AB 'completely and exactly'. Such objects are said to be congruent to one another. Another way of saying it is: The two objects are in congruence. So we see two terms: 'Congruent' and 'Congruence'. Their usage differs slightly. But both will denote the same condition:
• Line segments AB and PQ are congruent to one another
• Line segments AB and PQ are in congruence
We use the symbol '≡' for congruence of line segments. So we can write AB ≡ PQ
The Latin word 'congruo' means: 'I agree'. In day to day life, we see a lot of examples for congruence.
• We see 'text books of a particular subject' stacked in the book shelf in the store, ready for sale. Each of those books are congruent to one another.
• 'Cars of the same make' are congruent to one another.
• 'Bricks from the same mould' are congruent to one another.
What we saw in the fig.10.1 above was the 'congruence of line segments'.
■ If two line segments have the same length, they will be congruent to one another.
■ Conversely, If two line segments are congruent to one another, they have the same length.
The line segments XY and UV shown in fig.10.2 below, are not in congruence because, they do not have the same length.
If we make a trace copy of ∠XYZ and place it over ∠ABC, then XYZ will completely and exactly cover ABC. This is shown in fig.10.3(b). Note that, the lengths of the legs are immaterial as far as the angles are concerned. The measure of the angle is what we have to consider. ∠XYZ is able to cover ABC completely and exactly because both have the same measure, which is 65o. We can write:
• ∠ABC and ∠XYZ are congruent to one another. OR
• ∠ABC and XYZ are in congruence
Mathematically, this is written as: ∠ABC ≅ ∠XYZ
The angles ∠PQR and ∠MNO shown in fig.10.4 below are not in congruence because, they have different values: ∠PQR is 105o and ∠MNO is 25o
■ If two angles have the same measure, they are congruent to one another.
■ Conversely, If two angles are congruent to one another, then they have the same measure
In the next section we will discuss about Congruence of Triangles.
Fig.10.1(a) shows two line segments AB and PQ. These two segments are drawn in two different directions. So it is not easy to compare them.
 |
| Fig.10.1 |
• Line segments AB and PQ are congruent to one another
• Line segments AB and PQ are in congruence
We use the symbol '≡' for congruence of line segments. So we can write AB ≡ PQ
The Latin word 'congruo' means: 'I agree'. In day to day life, we see a lot of examples for congruence.
• We see 'text books of a particular subject' stacked in the book shelf in the store, ready for sale. Each of those books are congruent to one another.
• 'Cars of the same make' are congruent to one another.
• 'Bricks from the same mould' are congruent to one another.
What we saw in the fig.10.1 above was the 'congruence of line segments'.
■ If two line segments have the same length, they will be congruent to one another.
■ Conversely, If two line segments are congruent to one another, they have the same length.
The line segments XY and UV shown in fig.10.2 below, are not in congruence because, they do not have the same length.
 |
| Fig.10.2 |
Congruence of Angles
We will now see the congruence of angles. Fig.10.3(a) shows two angles: ∠ABC and ∠XYZ. They are oriented in different directions.
 |
| Fig.10.3 |
• ∠ABC and ∠XYZ are congruent to one another. OR
• ∠ABC and XYZ are in congruence
Mathematically, this is written as: ∠ABC ≅ ∠XYZ
The angles ∠PQR and ∠MNO shown in fig.10.4 below are not in congruence because, they have different values: ∠PQR is 105o and ∠MNO is 25o
 |
| Fig.10.4 |
■ If two angles have the same measure, they are congruent to one another.
■ Conversely, If two angles are congruent to one another, then they have the same measure
In the next section we will discuss about Congruence of Triangles.
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