# Triangles - Concept Points

Class 9 - Mathematics

1. A plane figure bounded by three line segments is called a triangle.

2. A triangle can be:
- equilateral (all three sides are equal)
- isosceles (any two sides are equal)
- scalene (all three sides are unequal)

3. A triangle in which one of the angles measures  90° is called a right-angled triangle or simply a right triangle.

4. The sum of the angles of a triangle is always 180°

5. Two figures are congruent, if they are of same shape and same size. Congruent triangles are exactly identical.

7. Congruent triangles have equal areas.

8. If two triangles △ABC and △XYZ are congruent under the correspondence A ↔ X, B ↔ Y and C ↔ Z, then symbolically, △ABC ≌ △XYZ

9. The abbreviation ‘c.p.c.t.’ for ‘corresponding parts of congruent triangles.

10. SAS Congruence Rule : If two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle, then the two triangles are congruent.

11.  ASA Congruence Rule : If two angles and the included side of one triangle are equal to two angles and the side of the other triangle, then the two triangles are congruent.

12. AAS Congruence Rule : If two angles and one side of one triangle are equal to two angles and the corresponding side of the other triangle, then the two triangles are congruent.

13. RHS congruence Rule : If in two right triangles, hypotenuse and one side of a triangle are equal to the hypotenuse and one side of other triangle, then the two triangles are congruent.

14. SSS Congruence rule : If three sides of one triangle are equal to the three sides of anoter triangle, then the two triangles are congruent.

15. Angles opposite to equal sides of a triangle are equal. (isosceles triangle)

16. Sides opposite to equal angles of a triangle are equal. (isoceles triangle)

17. Each angle of an equilateral triangle is 60°.

18. Of all the line segments that can be drawn to a given line from a point not lying on it, the perpendicular line segment is the shortest.

19. If two sides of a triangle are unequal,  the angle opposite to the longer side is greater

20. In a triangle, side opposite the greater angle is longer.

21. Sum of any two sides of a triangle is greater than the third side.

22. Difference between any two sides of a triangle is less than its third side.

23. In an isosceles triangle, the altitude from the vertex bisects the base.

24. The sum of three altitudes of a triangle is less than the sum of three sides of the triangle.

25. In a right triangle, the hypotenuse is the longest side.