Similarity
Similarity means having equal corresponding angles and proportional corresponding sides. For example, if shape A is doubled in size, then the resulting shape A′ will have corresponding angles that are the same as those in A and every side length of A′ will be double the corresponding side length of A.
The idea of similarity is different from congruence because congruent images must be the same size and shape while similar shapes only need to have the same shape. The notation for similarity is similar to the notation used to express congruence.
Instead of the symbol ≅, which is used to show congruence, use the symbol ∼ to show similarity. To show that triangle ABC and triangle XYZ are similar, use the notation ΔABC∼ΔXYZ.
Thales (640-540 BC) was a Greek mathematician, was the first who initiated and formulated the theoretical study of Geometry to make astronomy more exact science. Thales, who understood the concept of similarity, realized that because the sun hit objects in the same general area at the same angle, every object and its shadow create similar triangles. Thales compared the shadow of a stick with the shadow of the pyramid. Thales used the unit of a cubit to obtain these measurements. A cubit is a commonly used ancient unit and is equal to the distance from your elbow to your fingertips. In modern day, a cubit is considered to be 18 inches. Thales used a stick which was 6 cubits tall and stood it on the ground so that it was pointing directly upward. The shadow of the stick was 4 cubits long, and the shadow of the pyramid was 214 cubits long.
Similarity of triangles
For a given one-to-one correspondence between the vertices of two triangles, if
1) their corresponding angles are congruent and
2). Their corresponding sides are in proportion then the correspondence is known as similarity and the two triangles are said to be similar.
Two triangles are similar if and only if they have the three angles congruent , also, the equivalent sides must be proportional.
So in above example angles are equal and sides are in proportion.
Hence they are similar.
By I, Nguyenthephuc, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=2237929
I) ÐB @ ÐG, Ð A@ ÐE, ÐC @AÐF
ii. P = 2/ 3 , = 6 /9 = 2/3 , = 4/ 6 = 2/ 3 i.e., THESE TRIANGLES are similar triangles and are symbolically written as DABC DEFG,
Properties of the ratio of areas of triangles
Property-1
Ratio of the area of two triangles is equals to the ratio of the product of their corresponding base and height.
Property – 2
The ratio of areas of two triangles having equal base is equal to the ratio of their corresponding heights.
Property – 3
The ratio of areas of two triangles having equal height is equal to the ratio of their corresponding bases.
Property -4
Areas of two triangles having equal bases and equal heights are equal.
If two triangles are similar, then their corresponding angles are congruent and their corresponding sides are proportional. There are many theorems about triangles that you can prove using similar triangles.
the Basic Proportionality Theorem (BPT), also known as Thales Theorem.
Basic Proportionality Theorem (B.P.T)
Statement: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio. Converse of Basic Proportionality Theorem
The converse of Basic Proportionality theorem is also true.
Statement: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third.
Two figures are called similar if they have same shapes not necessarily the same size.
Properties of Similar Triangles: ABC.
1. Reflexivity: It means a triangle is similar to itself.DABC ~ D ABC.
2. Symmetry: If DDEF ~ D ABC. , then DABC ~ DDEF
3. Transitivity; ifD ABCDEF, DEF ~ PQR, then PQR ~ ABC
Angle Bisector Theorem
Statement: The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle.
The converse of the above theorem is also true.
Statement: If a line passes through one vertex of a triangle divides the base in the ratio of the other two sides, then it bisects the angle.
AAA test for similarity of triangles
For a given correspondence of vertices, when corresponding angles of
two triangles are congruent, then the two triangles are similar
when two angles of one triangle are congruent to two corresponding angles of another triangle then this condition is sufficient for similarity of two triangles. This condition is called AA test of similarity..
SAS test of similarity of triangles
For a given correspondence of vertices of two triangles, if two pairs of corresponding sides are in the same proportion and the angles between them are congruent, then the two triangles are similar.
Theorem of areas of similar triangles
Theorem : When two triangles are similar, the ratio of areas of those triangles is equal to the ratio of the squares of their corresponding sides