Show that the difference of any two sides of a triangle is less than the third side. GIVEN : `A B C` TO PROVE : (i) `A C-A BltB C` `B C-A CltA B` (iii) `B C-A BltA C` CONSTRUCTION : Take a point `D` on `A C` such that `A D=A Bdot` Join `B Ddot`

Show that the difference of any two sides of a triangle is less than the third side. Given : A A B C To Prove: (i) A C-A B

The sum of any two sides of a triangle is greater than the third side. GIVE : A A B C TO PROVE : A B+A C > B C ,A B+B C > A C and B C+A C > A B CONSTRUCTION : Produce side B A to D such that A D=A Cdot Join C Ddot

If two sides of a triangle are unequal, the longer side has greater angle opposite to it. GIVEN : A A B C in which A C > A Bdot TO PROVE : /_A B C >/_A C B CONSTRUCTION : Mark a point D on A C such that A B=A Ddot Joint B Ddot

Prove that any two sides of a triangle are together greater than twice the median drawn to the third side. GIVEN : A B C in which A D is a median. PROVE : A B+A C >2A D CONSTRUCTION : Produce A D to E such that A D=D Edot Join E Cdot

Show that the sum of the three altitudes of a triangle is less than the sum of three sides of the triangle. GIVEN : A A B C in which A D_|_B C ,B E_|_A C and C F_|_A Bdot PROVE : A D+B E+C F

In a right angled triangle, one acute angle is double the other. Prove that the hypotenuse is double the smallest side. GIVEN : A A B C in which /_B=90^0 and /_A C B=2/_C A B . to prove : A C=2B C CONSTRUCTION : Produce C B to D such that B D=C Bdot Join A Ddot

Show that a median of a triangle divides it into two triangles of equal area. GIVEN : A A B C in which A D is the median. TO PROVE : a r( A B D)=a r( A D C) CONSTRUCTION : Draw A L_|_B C

If D is the mid-point of the hypotenuse A C of a right triangle A B C , prove that B D=1/2A C . GIVEN : A A B C in which /_B=90^0 and D is the mid-point of A Cdot TO PROVE : B D=1/2A C CONSTRUCTION Produce B D to E so that B D=D Edot Join E Cdot

Angles opposite to two equal sides of a triangle are equal. GIVEN : A B C in which A B=A C TO PROVE : /_C=/_B CONSTRUCTION : Draw the bisector A D of /_A which meets B C in D Figure

In Figure, ABD is a triangle right-angled at A and A C_|_B D . Show that (i) A B^2=B C.B D (ii) A C^2=B C.D C (iii) A D^2=B D.C D