16. Triangle Properties of a Angle

 

TRIANGLE-PROPERTIES OF A TRIANGLE

Learn and Remember:

 1. Triangle : If A, B, C are three non-collinear points, then the figure formed by joining the segments AB, BC and CA is called a triangle ABC and written as ΔABC.

      More precisely, ΔABC is defined as the union of the segments AB, BC and CA, where A, B, C are non-collinear points.

(i) A triangle has three sides. e.g. sides of ΔABC are seg AB, seg BC and seg CA.

(ii) A triangle has three vertices. e.g. points A, B and C are the vertices of ΔABC.

(iii) A triangle has three angles. e.g. ∠ABC, ∠BCA and ∠CAB are the angles of ∠ABC.

2. The interior and the exterior of a triangle :

 A triangle divides the points in the plane of the triangle into three parts.

(i) Points on the triangle : Points L, M, N, P, Q, R.

(ii) Points in the interior of the triangle : Points A, B, C.

(iii) Points in the exterior of the triangle : Points T, D, S.

The triangular area : The triangle and its interior together form the triangular area.

3. Exterior angle of a triangle : An angle forming a linear pair with any angle of a triangle is called an exterior angle of that triangle. e.g. ∠ACP is an exterior angle of ΔABC. ∠BCE is also an exterior angle of ΔABC.

There are six exterior angles of a triangle. They form three pairs of congruent angles (each pair forms vertically opposite angles).

The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles.

e.g. in the figure,

m∠ACP = m∠CAB + m ∠ABC.

4. Triangles with respect to angles :

(i) An acute angled triangle : If all the three angles of a triangle are acute, the triangle is called an acute angled triangle.

(ii) An obtuse angled triangle : If one of the angles of a triangle is obtuse, the triangle is called an obtuse angled triangle.

(iii) A right angled triangle : If one of the angles of a triangle is a right angle, the triangle is called a right angled triangle. In a right angled triangle, the side opposite to the right angle is called 'hypotenuse'. Hypotenuse is the greatest side of the triangle. (Acute angles of a right angled triangle are complementary angles.)

5. The sum of the measures of the three angles of a triangle is 180°. Hence, a triangle cannot have more than one right angle or more than one obtuse angle.

6. Triangles with respect to sides :

(i) An equilateral triangle : A triangle having all the three sides congruent, is called an equilateral triangle. All the angles of an equilateral triangle are congruent, each having measure 60°.

(ii) An isosceles triangle : A triangle having two of its sides congruent is called an isosceles triangle. Angles opposite to the congruent sides of an isosceles triangle are congruent.

(iii) A scalene triangle : A triangle having no two sides congruent is called a scalene triangle. No two angles of a scalene triangle are congruent.

7. The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

8. The difference in the lengths of any two sides of a triangle is smaller than the length of the third side.

9. Types of triangles by both angles and sides :

  An acute angled             A right angled            An obtuse angled

isosceles triangle         isosceles triangle      isosceles triangle

The other three are (i) Scalene acute angled triangle (ii) Scalene right angled triangle (iii) Scalene obtuse angled triangle.

10. Concurrence in a triangle :

(1) Altitude :

(i) A perpendicular drawn from the vertex of a triangle to its opposite side is called an altitude. When an altitude is drawn on a side of a triangle, that side is known as the base, and the length of the altitude is called the height of the triangle.

(ii) A triangle has three altitudes.

e.g. in the figure, altitude AD is drawn from the vertex A on the side BC, altitude BE is drawn from the vertex B on the side AC and altitude CF is drawn from the vertex C on the side AB.

The altitudes of a triangle are concurrent.

The point of concurrence of the altitudes of a triangle is called the orthocentre.

(2) Perpendicular bisectors of the sides :

Perpendicular bisectors of the three sides of a triangle are concurrent. In the figure, C is the point of concurrence of the perpendicular bisectors of the sides. The point of concurrence of the perpendicular bisectors of the sides of a triangle is called the circumcentre.

(3) Angle bisectors :

The three angle bisectors of a triangle are concurrent. In the figure, I is the point of concurrence of the angle bisectors. The point of the angle bisectors of a triangle is called the incentre.

(4) Medians :

The three medians of a triangle are concurrent. In the figure, G is the point of concurrence of the medians. It is called the centroid.

Centroid divides the median in the ratio 2 : 1. AG:GP= 2:1.

 11. (1) The circumcircle of a triangle : To draw the circumcircle of a triangle, draw the perpendicular bisectors of any two sides of a triangle. The circumcentre of an acute angled triangle, a right angled triangle and an obtuse angled triangle lie in the interior of the triangle, on the midpoint of the hypotenuse, in the exterior of the triangle respectively.

(2) The incircle of a triangle : To draw the incircle of a triangle, draw the bisectors of any two angles of the triangle.

The incentre of any type of triangle lies in the interior of the triangle.

12. Construction of triangles : A triangle can be constructed if,

(i) all its three sides are known, or

(ii) the lengths of its two sides and the measure of the included angle are known, or

(iii) the measure of its any two angles and the included side are known, or

(iv) the lengths of the hypotenuse of a right angled triangle and one side are know.

13. Correspondence between the vertices of triangles :

Consider ΔABC and ΔPQR. A, B, C and P, Q, R are the vertices of the two triangles. A ↔ P means vertex A corresponds to vertex P and the vertex P corresponds to vertex A. Vertex A and vertex P are said to have one-one correspondence between them.

ABC ↔ PQR means A↔P, B↔Q, C↔R.

If ABC ↔ PQR, then the following are the corresponding angles and corresponding sides :

∠A↔∠P side AB ↔ side PQ

∠B↔∠Q and side BC ↔side QR

∠C↔∠R  side AC ↔ side PR

14. Congruent triangles :

If there is one-one correspondence between the vertices of two triangles, such that three sides of one triangle are congruent to the corresponding three sides of the other triangle and three angles of one triangle are congruent to the corresponding three angles of the other triangle, then the two triangles are said to be congruent,

e.g. if ABC ↔ PQR,

such that ∠A ≅ ∠P, ∠B≅∠Q ∠C ≅ ∠R

and side AB ≅ side PQ,

side AC ≅ side PR,

side BC≅ side QR,

then ΔABC = ΔPQR. .

Congruence of scalene triangles : If two scalene triangles are congruent, they are congruent only for one correspondence. • Congruence of isosceles triangles : If two isosceles triangles are congruent by one correspondence, they will be congruent by one other correspondence also. 

Congruence of equilateral triangles : If two equilateral triangles are congruent in any one correspondence, they are congruent in every other correspondence.

15. Tests for congruency of triangles :

(1) SAS (2) SSS (3) ASA (4) Hypotenuse-side.