Two figures are said to be similar, if they have the same shape but not necessarily the same size. If the angles of one triangle are equal to the angles of another triangle, then the triangles are said to be Equiangular. Equiangular triangles have the same shape but may have different sizes. Therefore, equiangular triangles are also called Similar Triangles.

Two triangles are similar if their corresponding angles are equal and corresponding sides are proportional.

 

AB/DE = BC/EF = CA/FD

∠A = ∠D

∠B = ∠E

∠C = ∠F

Properties

  1. The ratio of area of similar triangles is equal to the square of ratio of sides.
  2. If the ratio of corresponding sides equals to 1, then triangles become congruent.

Test for similarity of triangles

1. AAA Similarity Test: Three angles of one triangle are respectively equal to the three corresponding angles of the other triangle.

2. SAS Similarity Test: The ratio of two corresponding sides is equal and the angles containing the sides are equal.

3. SSS Similarity Test: The ratio of all the three corresponding side of the two triangles are equal.

Congruent Triangles

Two or more figures can be said congruent if and only if they all have same size and shape. Two triangles ABC and DEF are said to the congruent, if they are equal in all respects (equal in shape and size).

If ∠A =∠D, ∠B = ∠E, ∠C = ∠F

AB = DE, BC = EF; AC = DF

Then, ∆ABC ≡ ∆DEF or ∆ABC ≅ ∆DEF

Tests for Congruency

1. SAS Test: Two sides and the included angle of the first triangle are respectively equal to the two sides and included angle of the second triangle.

2. SSS Test: Three sides of one triangle are respectively equal to the three sides of the other triangle.

3. ASA Test: Two angles and one side of one triangle are respectively equal to the two angles and one side of the other triangle.

4. RHS Test: The hypotenuse and one side of a right-angled triangle are respectively equal to the hypotenuse and one side of another right-angled triangle.