using a ruler and a pair of compasses only construct
i)a quadrilateral ABCD where lineAB=6cm,BC=10cm,angleADC=135degrees
As usual, make a sketch.
It is obvious that our quad will not be unique.
I drew a base line AD
At D, I constructed a perpendicular.
I then bisected one of the 90° angles,
This gave me the obtuse angle of 135°
At this point we have to be sure that our compass arcs for
AB = 6 and
BC = 10 will intersect
On my sketch, I drew the diagonal AC
From the properties of lengths of sides of a triangle
AC > 6+10 or AC > 16
in the other triangle ADC,
AC > AD + DC
subtract these two inequalities:
0 > AD+DC - 16
AD + DC < 16
So now you can choose any lengths for AD and DC , as long as their
sum < 16
How about AD = 6, and DC = 5.
Now just set your compass at 6, draw an arc to establish AB,
set you compass at 10 , draw an arc to establish BC
Where they meet will be your B.
Now just join your lines. All conditions are met.