LEARN TESTS

Congruency and Similarity of Triangles

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Similar Triangles

Two triangles are said to be similar, if their corresponding angles are equal and corresponding sides are proportional. In other words, similar triangles have exactly the same shape, but not necessarily the same size.



There are three axioms or postulates which are helpful in determining if the given triangles are similar provided at least three quantities are known (except for AA postulate). The three postulates are stated as follows:

AAA Postulate (also called AA postulate)

If all the three angles of one triangle are equal to the corresponding angles of another triangle, then the two triangles are said to be similar.


Note: If two corresponding angles are equal for the given triangles, then the third angle should necessarily be equal in order to satisfy the rule “Sum of angles of a triangle is equal to 180o.”

SAS Postulate

If two sides of one triangle are proportional to two corresponding sides of the other triangle and the included angles are equal, they are said to be similar.

Here, AB/PQ = BC/QR
Therefore, the triangles ABC and PQR are similar.

SSS Postulate

If three sides of one triangle are proportional to the corresponding sides of the other, then they are said to be similar.

Therefore, the triangles ABC and PQR are similar.

Some properties of similar triangles

In two similar triangles, ratio of corresponding sides = ratio of corresponding heights = ratio of the lengths of corresponding medians = ratio of the lengths of the corresponding angular bisectors = ratio of inradii = ratio of circumradii = ratio of perimeters.

Ratio of areas is equal to the ratio of squares of corresponding sides in two similar triangles.

Congruent triangles

Two triangles are said to be congruent if their corresponding angles and corresponding sides are equal. Two congruent triangles can be considered equal in all respects including the sides, angles, lengths of medians, lengths of angular bisectors, circumradii, inradii, altitudes, perimeter and area.


There are three axioms or postulates which are helpful in determining if the given triangles are congruent provided at least three quantities are known. These axioms are stated as follows:

SSS Postulate

If all the three sides of one triangle are equal to the corresponding sides of another triangle, then the two triangles are said to be congruent.

SAS Postulate

If two sides of one triangle are equal to two corresponding sides of the other and the included angles are equal, they are said to be congruent.



ASA Postulate

If two angles and a side of one triangle are equal to the corresponding angles and side of the other, then they are said to be congruent.


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