In a triangle, prove that the greater angle has the longer side opposite to it

Asked by gpnkumar0 | 5th Sep, 2016, 08:53: PM

Expert Answer:

begin mathsize 12px style Given space we space have space straight a space increment ABC comma space such space that space angle ABC greater than angle ACB
To space prove space that colon space AC space greater than space AB
Construct space BD space on space AC space such space that space angle DBC equals angle ACB... left parenthesis Since space angle ABC greater than angle ACB comma space we space can space find space straight a space point space on space AC space that space is space straight D
such space that space angle DBC equals angle ACB right parenthesis end style
begin mathsize 12px style In space increment DBC comma space
angle DBC equals angle ACB space left parenthesis Construction right parenthesis
So comma space DB equals DC... left parenthesis straight i right parenthesis.... left parenthesis Sides space opposite space equal space angles space are space equal right parenthesis
Adding space AD thin space to space both space sides comma space we space get
DB plus AD equals DC plus AD
rightwards double arrow DB plus AD equals AC... left parenthesis ii right parenthesis
In space increment ADB comma space we space have
DB plus AD greater than AB... left parenthesis iii right parenthesis... left parenthesis Sum space of space any space two space sides space of space straight a space triangle space is space greater space than space the space third space side right parenthesis
From space left parenthesis ii right parenthesis space and space left parenthesis iii right parenthesis comma space we space get
AC greater than AB
Hence space proved. end style

Answered by Rebecca Fernandes | 5th Sep, 2016, 10:53: PM