If an angle of one triangle is congruent to an angle of a se

If an angle of one triangle is congruent to an angle of a second triangle and the lengths of any two sides in the first and second triangles are proportional, then the triangles are similar.

Solution

The statement is

\" If an angle of a triangle is congruent to an angle of another triangle and the lengths of any two sides in the first triangle is proportional to lengths of two sides of another triangle then the two triangles are similar\"

The above statement is false.

Now why is it false ??

In order to justify our conclusion we need to revisit the SAS theorum of similar triangles.

What does it say?

it states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then these two triangles are congruent.

The clue is provided in this theorum only.

It talks about an angle and two sides which subtend that angle in the triangle and NOT any two sides of a triangle.

While in our statement it says that the an angle in two triangles are congurent and any two sides of one are proportional to two sides of other. Clearly it doesnt mention that the sides must need to be those sides which subtend the angle in question ( i.s sides are NOT the nicluded sides)

Hence the statement is FALSE.

The correct stement should have been as follows

\" If an angle of a triangle is congruent to an angle of another triangle and the lengths of sides subtending that angle in the first triangle is proportional to lengths of sides subtending the same angle in another triangle then the two triangles are similar\"