Geometry

A triangle is a closed figure formed by three line segments. It has three vertices, three sides and three angles.

                                                                                    A

 

 

                                            B                                                             C

 

VERTICES:-  A , B , C

SIDES:-  AB , BC , CA

ANGLES:-  ∠CAB , ∠ABC , ∠BCA

Ř In any triangle drawn by joining three non-collinear points, the sum of the lengths of any two sides is greater than the length of the third side. This property is called as Triangle inequality.

 

To verify this property, let us consider three types of triangles based on angles.

For each of these triangles the following statements are true:-

1. a + b > c

2.b + c  > a

3. c + a > b

This property is also true for three types of triangles based on sides.

Application Of Angle Sum Property Of Triangle:-

Theorem : Sum of all angles in a triangle is 180°.

Given: A triangle ABC, where ∠A =  x , ∠B =  y and ∠C = z .

To prove :-  x + y + z = 180.                                                    

Proof :-  

·        Extend BC to D and draw a line CE, parallel to AB.   

·        Now CE makes two angles, ∠ACE and ∠ECD . Let it be u and v respectively.

·        Now z, u, v are the angles formed at a point on a straight line. Therefore z + u + v = 180 .

·        Since AB and CE are parallel and DB is a transversal,  v = y (corresponding angles).

·        Again AB and CE are parallel lines and AC is a transversal,  u = x (alternate angles).

Also  z +u + v = 180  .

·        Hence, by replacing u as x and v as y, we get x + y + z = 180 .Hence the sum of all three angles in a triangle is 180.

EXAMPLES:-

EXTERIOR ANGLES:-

 


       Interior opposite angles

 


                                                                                                            Exterior angle

 

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

To prove:-  f=b+c,  e=a+c and  d=a+b

Proof :-

·        In ΔABC, consider, the angles at A, B and C as a, b and c respectively and take exterior angles at A, B and C as d,e,f respectively.

·        a + f = 180  [linear pair of angles are supplementary]

This gives, f = 180° – a                                                           … (1)

·         Now, a + b + c = 180° [Sum of 3 angles in a triangle is 180°]

 This gives, b + c = 180° – a                                                               …(2)

·        From (1) and (2),  x and b + c both are equal.  Therefore, f = b + c.

 


Theorem :- The sum of exterior angles of a triangle is 360° .

To Prove:-   x + y + z = 360°

Proof :-

EXAMPLES

Congruency Of Triangles

Any pair or set of are said to be congruent if they exactly match with the other in size and shape.

Congruence of Line Segments

                                                    A                                         B

                                                               C                                          D

 

The line segments  and   have the same length. If we use the superposition method   and   will match each other. Hence the line segments are congruent and we can write it as       .

Congruence of Angles

Congruence of Triangles

If all the sides and all the angles of one triangle are equal to the corresponding sides and angles of another triangle, then the two triangles are congruent to each other.

 

Conditions for Triangles to be Congruent:

To construct a triangle, we need only three measures. Those three measures can be any of the following:

1.     The lengths of all three sides.

2.     The lengths of two sides and the angle included between those two sides.

3.     Two angles and the length of the side included by angles.

1. Side–Side–Side congruence criterion (SSS) - The lengths of all three sides are given

2. Side-Angle-Side congruence criterion (SAS) - The lengths of two sides and the angle included between the two sides are given.

3.     Angle – Side – Angle congruence criterion (ASA) - Two angles and the length of a side included by the angles are given.

Hypotenuse

The side which is opposite to the right angle is the largest side called Hypotenuse.

4.     Right Angle – Hypotenuse – Side congruence criterion (RHS)

HYPOTENUSE

·        In these two triangles right angle is common. And if we are given the sides making right angles then we can use the SAS criterion to check the congruency of the triangles.

·         Or if we are given one side containing right angle and hypotenuse, then we can have new criterion, if the hypotenuse and one side of a right angled triangle is equal to the hypotenuse and one side of another right angled triangle then the two right angled triangles are congruent. This is called Right angle – Hypotenuse – Side criterion.