06. TRIANGLES

Triangle  : A triangle is a simple closed curve made of three line segments. It has three vertices, three sides and three angles.

An exterior angle of a triangle is equal to the sum of its interior opposite angles.

The sum of the  three angles of a triangle is  180 degrees.

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

In any triangle, the side opposite to the larger (greater) angle is longer.

Median of the triangle : The line segment joining a vertex of a triangle to the mid point of its opposite side is called a median of the triangle. A triangle has 3 medians.

Centroid of a Triangle :The point of intersection of the medians of the three sides of the Triangle is the centroid of that Triangle. It will always inside the Triangle.

Incenter of a Triangle : The point of intersection of the angle bisectors of the three angles of the Triangle is called the incenter of that Triangle. 

Circumcenter of the Triangle : The point of intersection of the perpendicular bisectors of the three vertices of the Triangle is called the circumcenter of that Triangle.

Orthocenter : The point of intersection of the altitudes of the Triangle is the orthocenter of that Triangle.

Altitude of the triangle : The perpendicular line segment from a vertex of a triangle to its opposite side is called an altitude of the triangle. A triangle has 3 altitudes.

Types of triangle

1. Based on Sides:

Scalene Triangle : A triangle in which all the three sides are unequal in length is called a scalene triangle.

Isosceles Triangle : A triangle in which two sides are of equal in lengths is called an isosceles triangle.

In an isosceles triangle:

(i) two sides have same length.

(ii) base angles opposite to the equal sides are equal

Equilateral Triangle : A triangle in which all the thre sides  are equal is called an equilateral triangle. In an equilateral triangle, each angle has measure 60°

Based on Angles

right angled triangle : A right angled triangle is a triangle in which one of the angles is 90°.

In a right-angled triangle, 

The side opposite to the right angle is called the hypotenuse

The other two sides are known as the legs of the right-angled triangle.

The square on the hypotenuse = sum of the squares on the legs.

Obtuse angled triangle. : A triangle in wich any one angle is greater than 90°, is called an obtuse angled triangle.

Acute angled triangle : A triangle in which each angle is less than 90°, is called an acute angled triangle

congruent figures. : Two figures having the same shape and the same size, are called congruent figures.

Similar figures. : Two figures having the same shape but not necessarily the same size are called similar figures.

Two circles of the same radii are congruent

Two squares of the same sides are congruent.

Two equilateral triangles with the same side lengths are congruent.

All congruent figures are similar but the similar figures need not be congruent.

Two polygons of the same number of sides are similar, if (i) all the corresponding angles are equal and (ii) all the corresponding sides are in the same ratio/ proportion.

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

SAS congruence rule : Two triangles are congruent if two sides and the included angle of one triangle are equal to the two sides and the included angle of the other triangle.

ASA congruence rule: Two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of other triangle.

 AAS Congruence Rul: Two triangles are congruent if any two pairs of angles and one pair of corresponding sides are equal. 

SSS congruence rule : If three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent.  

RHS congruence rule : If in two right triangles the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle, then the two triangles are congruent. Note that RHS stands for Right angle - Hypotenuse - Side.

Basic Proportionality Theorem:

Theorem 6.1 : If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

Theorem 6.2 : If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

        AD  ₌  AE .
        DB     EC

Theorem 6.3 : If in two triangles, corresponding angles are equal, then their

corresponding sides are in the same ratio (or proportion) and hence the two

triangles are similar.

Theorem 6.4 : If in two triangles, sides of one triangle are proportional to

(i.e., in the same ratio of ) the sides of the other triangle, then their corresponding

angles are equal and hence the two triangles are similiar.

6.5 : If one angle of a triangle is equal to one angle of the other

triangle and the sides including these angles are proportional, then the two

triangles are similar.

EXERCISE 6.1

1. Fill in the blanks using the correct word given in brackets :
   (i) All circles are similar.      (congruent, similar)
   (ii) All squares are similar. (similar, congruent)
   (iii) All equilateral triangles are similar . (isosceles, equilateral)
   (iv) Two polygons of the same number of sides are similar, if (a) their corresponding angles are equal and (b) their corresponding sides are proportional . (equal, proportional)
2.  Give two different examples of pair of

   (i) similar figures. 

   (a) Any two rectangles

   (b) Any two squares

   (ii) non-similar figures.

   (a) A scalene and an equilateral triangle

   (b) An equilateral triangle and a right angled triangle
3. State whether the following quadrilaterals are similar or not:
   
   From the given two figures, we can see their corresponding angles are different or unequal. Therefore, they are not similar.

EXERCISE 6.2

1. In Fig. 6.17, (i) and (ii), DE || BC. Find EC in (i) and AD in (ii).
   
   Fig (i) by using Basic proportionality theorem
   AD  ₌  AE
   DB     EC
   1.5  ₌   1  
    3      EC
   EC ₌   3 
           1.5
   EC ₌ 2 cm
   Fig (ii) According to Basic Proportionality Theorem
   AD  ₌  AE
   DB     EC
   AD  ₌   1.8  
   7.2       5.4
   AD ₌  1.8x 7.2
               5.4
   EC ₌ 2.4 cm
2. E and F are points on the sides PQ and PR respectively of a △ PQR. For each of the following cases, state whether EF || QR :
   (i) PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm
   
   by using Basic proportionality theorem
   
   
   Therefore , EF is not parallel to QR.
   (ii) PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm
   
   by using Basic proportionality theorem
   
   Therefore , EF is parallel to QR.
   (iii) PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36
   Here, PQ = 1.28 cm, PR=2.56 cm, PE = 0.18, PF= 0.36 cm
   
   EQ =  PQ - PE
   EQ = 1.28 - 0.18 = 1.10 cm
   And
   FR = PR - PF
   FR = 2.56 - 0.36 =2.20cm
   
3. In Fig. 6.18, if LM || CB and LN || CD, prove that 

   

   
   
4. In Fig. 6.19, DE || AC and DF || AE. Prove that  
   
   
   
5. In Fig. 6.20, DE || OQ and DF || OR. Show that EF || QR.
   
   
   
6. In Fig. 6.21, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR.
   
   
7. Using Theorem 6.1, prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side.
   
   
8. Using Theorem 6.2, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side.
   
9. ABCD is a trapezium in which AB || DC and its diagonals intersect each other at the point O. Show that
                                   
                                        
   
10. The diagonals of a quadrilateral ABCD intersect each other at the point O such that
    
     Show that ABCD is a trapezium.