Triangle: Triangle is a simple closed figure consisting of three line segments. In fig. 1.1 the line segments BC, CA and AB form a triangle and it is named triangle ABC. The line segments BC, CA and AB are called the sides of the triangle.

Types of Triangles: We can classify triangles by

i. Considering the lengths of their sides

ii. Considering the magnitudes of their angles.

TRIANGLES

a. Let us consider the following triangles in fig 1.2

a. A triangle in which no two sides are equal is called a scalene triangle. Fig 2.(a)

b. A triangle in which two sides are equal is called an isosceles triangle.

c. A triangle in which all the sides are equal is called an equilateral triangle. In fig 1. 2 (c)

In (a), (b) ,(c) above we have taken lengths of sides into consideration in naming the triangle.

Types of Triangles

a. A triangle in which all the three angles are a cute is called an acute triangle. Fig 1.3 (a)

b. A triangle in which one of the angles in an obtuse angle is called an obtuse triangle. Fig 1.3 (b)

c. A triangle in which one of the angles is a right angle is called a right (angled) triangle. Fig 1.3 (c)

II. Now let us consider the following triangles

1. Two plane figures are said to be congruent if they have same size and same shape.

2. The sum of the angles of a triangle is 1800

3. Two congruent figures can be made to coincide with each other by the method of superposition.

4. If a side and two angles of one triangle are equal to a side and two corresponding angles of another triangle, then the two triangles are congruent (The ASA property).

5. If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, than the two triangles are congruent (The SAS property)

6. If three sides of a triangle are respectively equal to the three sides of another triangle, then the two triangles are congruent.

7. If the hypotenuse and a side (by) of a right triangle are equal to the hypotenuse and a side of another right triangle, then the two triangles are congruent (The RHS property)

Properties of triangles:

8. When two triangles are congruent by any of the four properties stated in (3),(4),(5),(6) above the remaining corresponding parts of the two triangles become equal.

9. Two line segments are congruent if they have the same length.

10.Two circles are congruent if they have the same radius.

11. If two sides and the included angles of a triangle are respectively equal to two sides and the included angle of another triangle, then two triangles are congruent.

12. If two angles and a side of one triangle are respectively equal to two angles and the corresponding side of another triangle. Then the two triangles are congruent.

13. If the sides of a triangle are equal to the three sides of another triangle. Then the two triangles are congruent.

14. If the hypotenuse and a side of a right angled triangle are equal to the hypotenuse and a side of another right angled triangle, then the two triangles are congruent.

15.The perpendicular bisectors of the sides of a triangle are concurrent.

16. The point of concurrence of the perpendicular bisectors of a triangle is equidistant from the vertices of the triangle.

17.The circle passing through the vertices of a triangle is called the CIRCUMCIRCLE. It is centre is called the CIRCUMCENTRE and its radius is called the CIRCUMRADIUS. The CIRCUMCENTRE is equidistant from the vertices of the triangle.

18.The internal bisectors of the angles of a triangle are concurrent.

19.The point of concurrence of the internal bisectors of the angles of triangle is equidistant from the sides of the triangles.

20.The external bisectors of any two angles of a triangle and the internal bisector of the third angle of the triangle are concurrent.

21.The point of concurrence of the external bisectors of two angles of a triangle and the internal bisector of the third angle is equidistant from the sides of the triangle.

23.The medians of a triangle are concurrent.

24. Altitudes of a triangle are the perpendiculars drawn from the vertices of a triangle to the opposite sides.

25.The orthocenter is the point of concurrence of the altitudes of a triangle.

PREPARED BY :

NAMES: N.S.V.Prasad, S.K.Safiya

Class: IX class Telugu Medium

Script: Maths TextBook (VII class)

Pictures : S.K.Safiya