Q. 55.0( 1 Vote )

Explain with reasons, whether the following statements are true or false:

(1) AAA criterion of similarity of triangles cannot be the criterion for congruence of triangles.

(2) SAS criterion for congruence of triangles cannot be a criterion for similarity of triangles.

(3) Two congruent triangles have the same area.

(4) Two similar triangles always have the same area.

(5) Area of similar triangles are proportional to the squares of measures of their corresponding angles.

Answer :

(1). Similarity of triangles and congruence of triangles are closely related, but has a fine line of difference.


Similar triangles are the same shape, but not necessarily the same size. The triangles are congruent if, in addition to this, their corresponding sides are of equal length.


By AAA criterion of similarity of triangles, we get the shapes of triangles equal but, for congruence of triangles their sizes must be equal too.


AAA criterion of similarity of triangles cannot be the criterion for congruence of triangles.


Hence, the statement is true.


(2). For congruence of triangles, the corresponding sides are of equal length while for similarity of triangles, measure of the corresponding sides of two triangles must be proportional.


SAS criterion for congruence of triangles not only makes sure that the triangles have equal length of corresponding sides but also makes sure that the triangles have same shape.


SAS criterion for congruence of triangles can be a criterion for similarity of triangles.


Hence, the statement is false.


(3). Congruency of triangles ensure that the triangles have same shape as well as equal lengths of corresponding sides.


The shape and size of two congruent triangles are equal.


Two congruent triangles have the same area.


Hence, the statement is true.


(4). Similarity of triangles indicate that they have the same shape but their sizes are not same.


Area of triangles are equal when their size are equal, shape of triangle cannot determine its area.


Two similar triangles doesn’t always have the same area.


Hence, the statement is false.


(5). Area of similar triangles are proportional to the squares of their corresponding sides.


Since, the measure of corresponding angles of similar triangles are always equal.


Area of similar triangles are not proportional to the squares of their corresponding angles.


Hence, the statement is false.


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