We often come across two triangles that share the same proportions, i.e. the angles are alike though their sizes differ. These are called similar triangles. The simplest method of illustrating two similar triangles is first to draw a triangle, and then to draw a straight line so that we now have a smaller triangle within the large one.
For triangles to be similar:
– All corresponding angles are equal,
– All corresponding sides have the same ratio
If you notice carefully, then you will notice that the corresponding sides will face corresponding angles. For example, the sides that face the angles with two arcs are corresponding.
In similar triangles, corresponding sides are always in the same ratio.
Triangles R and S are similar. The equal angles are marked with the same numbers of arcs.
How to find Similar Triangles:
The two triangles are similar if:
– All their angles are equal
– Corresponding sides are in the same ratio
But, it is not necessary to know all the sides of the triangles. There are theorems by which can identify whether the triangles are similar or not.
AA means “angle-angle” similarity criterion. This means the triangles have two of their angles equal.
If two triangles have two of their angles equal, then the triangles are similar.
If angles are equal, then the third angle is compulsorily equal, because angles of a triangle always add to make 180 degrees.
So AA could also be called AAA (because when two angles are equal, all three angles must be equal).
SAS means “side-angle-side” similarity criterion. It means that
the ratio between two sides is the same as the ratio between another two sides
and the included angles are also equal.
If two triangles have two pairs of sides in the same ratio and the included angles are also equal, then the triangles are similar.
SSS means “side-side-side” similarity criterion.
If two triangles have three pairs of sides in the same ratio, then the triangles are similar.
BASIC PROPORTIONALITY THEOREM
If a line is drawn parallel to one side of a triangle to intersect the other sides in distinct points, the other two sides are divided in the same ratio.
Converse of Basic Proportionality Theorem:
- If in two triangles, the corresponding angles are equal, their corresponding sides are proportional and the triangles are similar.
- If one angle of a triangle is equal to one angle of other triangle and the sides including these angles are proportional, the triangles are similar.
- If a perpendicular drawn from the vertex of the right angle of a right triangle to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle and to each other.
AREA OF SIMILAR TRIANGLES
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Converse of Pythagoras Theorem:
In a triangle, if the square of one side is equal to the sum of the squares of the other two sides then the angle opposite to the first side is a right angle.