Triangles Class 9 Notes Maths

Triangles Class 9 Notes CBSE for Maths

Polygon

A closed plane figure bounded by line segments is called a polygon.

The line segments are called its sides and the points of intersection of consecutive sides are called its vertices. An angle formed by two consecutive sides of a polygon is called an interior angle or simply an angle of the polygon.

A polygon is named according to the number of sides it has :

No.of sides	3	4	5	6	7	8	9	10
Name	Triangle	Quadrilateral	Pentagon	Hexagon	Heptagon	Octagon	Nenagon	Decagon

In general, a polygon having n sides is called n-gon. Then a polygon having 18 sides is called 18-gon.

Diagonal of Polygon

Line segment joining any two non-consecutive vertices of a polygon is called its diagonal.

Convex Polygon

If  all the interior angles of a polygon are less than 180°, it is called a convex polygon.

Figure

Concave Polygon

If one or more interior angles of a polygon are greater than 180° i.e. reflex, it is called a concave polygon.

Figure

Regular Polygon

A polygon is called a regular polygon if all its sides have equal length and all the angles are equal.

Figure

Remarks

(1)  The sum of the interior angles of a convex polygon of n sides is (2n – 4) right angles.

(2)  Each interior angle of a n-sided regular polygon is \frac{{\left( {2n - 4} \right)\,{\bf{right}}\,\,{\bf{angles}}}}{n}.

(3)  Each exterior angle of a regular polygon of n sides =\frac{4}{n}{\bf{right}}\,\,{\bf{angles}} = \left( {\frac{{{{360}^0}}}{n}} \right) .

(4)  If a polygon has n sides, then the number of diagonals of the polygon =  \frac{{n\left( {n - 1} \right)}}{2} - n or \frac{{n\left( {n - 3} \right)}}{2}

Triangle

A plane figure bounded by three lines in a plane is called a triangle. Every triangle has three sides and three angles. If ABC is any triangle  then  AB,  BC  &  CA  are  three  sides  and  ∠A, ∠B and ∠C are three angles.

Types of triangles              

   

Figure

On the basis of measure of sides we have three types of triangles :

1. Scalene triangle – A triangle whose no two sides are equal is called a scalene triangle.
2. Isosceles triangle – A triangle having two sides equal is called an isosceles triangle.
3. Equilateral triangle – A triangle in which all sides are equal is called an equilateral triangle.

On the basis of measure of angles we have three types of triangles:

1. Right triangle – A triangle in which any one angle is right angle is called a right triangle.
2. Acute triangle – A triangle in which all angles are acute is called an acute triangle.
3. Obtuse triangle – A triangle in which any one angle is obtuse is called an obtuse triangle.

Exterior angle of a triangle                                                                                 

If the side BC of a Δ ABC is produced to form ray BD, then ∠ACD is called an

exterior angle of Δ ABC at C.

Exterior angle theorem

If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles.

In a triangle ABC, D is a point on BC produced forming exterior angle ∠4.

Then  ∠4 = ∠1 + ∠2

Figure

Congruent Triangles

Congruent Figures

Two figures are called congruent if they have same shape and same size.

In other words, two figures are called congruent if trace copy of one figure is superposed on other so as to cover it completely and exactly.

Congruent Triangles

Two triangles are congruent if and only if one of them can be made to superpose on the other, so as to cover it exactly.

Thus, congruent triangles are exactly identical.

(a)                                             (b)

Figure

             If two triangles Δ ABC & Δ DEF are congruent then there exist a one to one correspondence between their vertices & sides. We get following six equalities

∠A = ∠D,  ∠B = ∠E, ∠C = ∠F

AB = DE, BC= EF, AC = DF.

If two ΔABC & ΔDEF are congruent under A « D, B « E, C « F one to one correspondence then we write  Δ ABC ≅ Δ DEF.

We can not write as Δ ABC ≅ Δ DFE or Δ ABC ≅ Δ EDF or in other forms because Δ ABC ≅ Δ DFE have following one-one correspondence

A « D, B « e, C « f. Hence we can say that “two triangles are congruent if and only if there exists a one-one correspondence between

their vertices such that the corresponding sides and the corresponding angles of the two triangles are equal.

Note:    (i) If ΔABC ≅ ΔDEF then we have : ∠A = ∠D, ∠B = ∠E, ∠C = ∠F; and AB = DE, BC = EF and AC = DF

(ii) If ΔABC ≅ ΔEDF then we have; ∠A = ∠E, ∠B = ∠D, ∠C = ∠F and AB = ED, BC = DF and AC = EF

Congruence relation in the set of all triangles

By the definition of congruence of two triangles, we have following results.

1. Every triangle is congruent to itself i.e. Δ ABC ≅Δ ABC.
2. If Δ ABC ≅ Δ DEF then Δ DEF ≅ Δ ABC.
3. If Δ ABC ≅ Δ DEF and Δ DEF ≅ Δ PQR then Δ ABC ≅ ΔPQR.

Theorem 1.

            Two triangles are congruent if two sides and the included angle of one are equal to the corresponding sides and the included angle of the other triangle.

In  two triangles ABC and DEF such that AB = DE, AC = DF and ∠A = ∠D.

Figure

            Then      ΔABC ≅ ΔDEF.

            Note :

            It shall be noted that in SAS criterion the equality of included angles is very essential. If two sides and one angle (not included between the two sides) of one triangle are equal to two sides and one angle of the other triangle, then the triangles need not be congruent. So, the equal angle should be the angle included between the sides.

Theorem 2.

Two triangles are congruent if two angles and the included side of one triangle are equal to the corresponding two angles and the included side of the other triangle.

Figure

            In two Δs ABC and DEF such that

∠B = ∠E,    ∠C = ∠F           and       BC = EF

            Then                             Δ ABC ≅ ΔDEF

Theorem 3.

            If any two angles and a non-included side of one triangle are equal to the corresponding angles and side of another triangles, then the two triangles  are congruent.

Figure

            In two Δs ABC and DEF such that

∠A = ∠D,  ∠B = ∠E,  BC = EF

            Then           ΔABC ≅ ΔDEF

Theorem 4.

            Angles opposite to two equal sides of a triangle are equal.

In ΔABC in which AB = AC.

            Then              ∠B = ∠C.

       

Figure

Theorem 5.

            If two angles of a triangle are equal, then sides opposite to them are also equal.

            In a ΔABC in which ∠B = ∠C.

            Then             AB = AC

Figure

Theorem 6.   

                         

Figure

            If the bisector of the vertical angle of a triangle bisects the base of the triangle, then the triangle is isosceles.

In a ΔABC in which AD is the bisector of ∠A meeting BC in D such that BD = DC.

Then ΔABC is an isosceles triangle.

Theorem 1.

            Two triangles are congruent if the three sides of one triangle are equal to the corresponding three sides of the other triangle.

Figure

In two ΔsABC and DEF such that AB = DE, BC = EF and AC = DF.

            Then            ΔABC ≅ ΔDEF

Theorem 2.

Two right triangles are congruent if the hypotenuse and one side of one triangle are respectively equal to the hypotenuse and one  corresponding side of the other triangle.

Figure

In two right triangles ABC and DEF in which ∠B = ∠E = 90°, AC = DF, BC = EF.

            Then                    ΔABC ≅ ΔDEF

Inequalities in Triangles 

Introduction : We know that if in a triangle any two sides are equal then their opposite angles are also equal and if any two angles are equal then their opposite sides are also equal. Now if in a triangle any two sides are not equal then what will be the relation between their opposite angles. A relation in which two quantities are not equal is called an inequality relation i.e. if a ≠ b then either a > b or a < b these relations are called inequalities. Now we will learn some inequalities in a triangle.

Theorem 1

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

In a ΔABC in which AC > AB.

Figure

Then              ∠ABC > ∠ACB

Theorem 2                                                                                                                    

            (Converse of Theorem 1) In a triangle the greater angle has the longer side opposite to it.

Figure

In a ΔABC in which ∠ABC > ∠ACB.

            Then                  AC > AB.

Theorem 3

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

In a ΔABC, AB + AC > BC, AB + BC > AC and BC + AC > AB.

Figure

Theorem 4

            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.

Figure

In a straight line l and a point P not lying on l. PM ⊥ l and

N is any point on l other than M.

Then                                 PM < PN

 

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