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| Congruent Triangles |
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| When two triangles have (i) the same shape, and (ii) the same size, then they are said to be congruent triangles. |
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Symbol for congruency is  |
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Draw  with AB = 4 cm, AC = 5 cm, and  . |
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Also  with DE = 4 cm, DF = 5 cm and  |
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| Draw these triangles on a tracing paper. Now place fig(ii) on fig.(i), so that D is placed on A, DE along AB and F on the same side as C. |
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It is noticed that  coincides with   exactly fits on  |
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| If two triangles have two sides of one equal to two sides of the other, each to each, and the angles included by these sides are equal, then the triangles are congruent. |
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| The equal angles must be formed by the two pairs of equal sides. |
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Also draw  and EF = 5 cm. |
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| Draw these triangles on a tracing paper. |
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Place Fig.(ii) on Fig.(i), EF along BC, [  EF = BC ]. |
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Then we notice that  coincides with  ,  |
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| If two triangles have two angles of one equal to two angles of the other, each to each, also one side of the one equal to the corresponding side of the other, then the triangles are congruent. |
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| This condition is also named as (ASA) or (SAA), but the proper name is two angles and the corresponding sides (AA cor. S). |
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The two triangles ABC and DEF given below are not congruent although  and AB = DF. This is because the sides AB and DF are not corresponding sides. |
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| If two triangles have three sides of one, equal to the corresponding sides of the other, then the triangles are congruent. |
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| In the figure, the two triangles are congruent since the three sides of one triangle are equal to the corresponding three sides of the other. |
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| This condition is a modification of SAS Axiom. Here the two triangles: |
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| (1) must be right angled. |
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| (2) Their hypotenuses must be equal |
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| (3) Any other side of one must be equal to any one of the remaining two sides of the other. |
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| If two right angled triangles have their hypotenuses equal, and one side of one equal to one side of the other, then the triangles are congruent. |
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