CHAPTER 6
TRIANGLES
1. Two figures having the same shape and same size are congruent figures.
2. Two figures having the same shape but not the same size are similar figures.
3. All congruent figures are similar but all similar figures need not be congruent.
4. Two polygons having the same number of sides will be similar if the corresponding angles of the two polygons are equal and the corresponding sides are proportional.
5. Basic Proportionality Theorem : If a line is drawn parallel to one side of a triangle to intersect the other two sides in two distinct points, then the other two sides are divided in the same ratio.
$$\mathrm{PQ} \| \mathrm{BC} ; \frac{A P}{P B}=\frac{A Q}{Q C}$$
6. Converse of a Basic Proportionality Theorem: If a line divides any two sides of a triangle in the same ratio, then the
line is parallel to the third side.
SIMILAR TRIANGLES
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| (A) | AAA Similarity: If the corresponding angles of two triangles are equal, then their corresponding sides are proportional. Hence, the two triangles are similar. |
| (B) | AA Similarity: If two angles of one triangle are equal to the corresponding two angles of another triangle, then the two triangles are similar. |
| (C) | SSS Similarity: If the corresponding sides of two triangles are proportional (then the corresponding angles are equal), the two triangles are similar. |
| (D) | SAS Similarity: If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio, then the two triangles are similar. |
| (E) | RHS Similarity: In two right-angled triangles, if the hypotenuse and one corresponding side are proportional, then the two triangles are similar. |
IMPORTANT THEOREMS
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| 8. | Pythagoras Theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. |
| 9. | Converse of Pythagoras Theorem: If in a triangle, 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. |
| 10. | The line joining the mid-points of two sides of a triangle is parallel to the third side and is half of the third side. |
| 11. | The diagonals of a trapezium divide each other proportionally. |