Triangle Class 10th
Congruent – Polygons in which all sides or angles are equal to those of another polygon are called congruent polygons.
congruent-
Similar figures – Polygons that are similar in shape but differ in size.
Note – All congruent polygons are similar, but not all similar polygons are congruent.
All circles, equilateral triangles, and squares are similar figures.
Equiangular triangles – If the three angles of one triangle are equal to the three angles of another triangle, they are called equiangular triangles.
All equiangular triangles are similar.
Similar triangles – Triangles whose corresponding angles are equal and whose corresponding sides are in the same ratio (i.e., the sides are proportional).
Q. 1. Fill in the blanks –
1. All circles are similar. (Congruent / Similar)
2. All squares are similar. (Congruent / Similar)
3. All equilateral triangles are similar. (Isosceles / Equilateral)
4. Two polygons with the same number of sides are similar if their corresponding angles are equal and their corresponding sides are proportional. (Equal / Proportional)
Q. 2. Give two different examples for each of the following pairs –
1. Examples of similar figures –
- Any two circles (with different radii).
- Any two squares (with different side lengths).
2. Figures that are not similar (non-similar figures) –
- An equilateral triangle and a scalene triangle.
- A rectangle and a rhombus.
Q. 3. State whether the following quadrilaterals are similar or not –
Ans. No, these quadrilaterals are not similar.
Thales’ Theorem / Law of Basic Proportionality.
Theorem 6.1 – If a line is drawn parallel to one side of a triangle to intersect the other two sides at different points, then these other two sides are divided in the same ratio.
Theorem 6.2 – If a line divides two sides of a triangle in the same ratio, then it is parallel to the third side.
Theorem 6.3 – If in two triangles, the corresponding angles are equal, then their corresponding sides are in the same ratio (proportional) and hence these triangles are similar.
Criteria for similarity of triangles:
1. Side-Side-Side: If the ratio of the three sides of any two triangles is equal, then the triangles are similar.
2. Angle – Angle – Angle – If the corresponding angles of any two triangles are equal then those triangles are similar triangles.
3. Side – Angle – Side – If the ratio of two sides of any two triangles is equal and the corresponding angle between them is equal to the corresponding angle of the other triangle, then those triangles are similar triangles.
4. Angle – Angle– If the two corresponding angles of any two triangles are equal then those triangles are similar triangles.
