Question

St abd uv are intersecting at a such that the lengths of os ot ou ov in (cm) are 4.2 6.3 1.2 and 1.8 prove su parallel vt

Answer 1

In Δ SOU and Δ TOV we have
SO = OT [∴ O is the mid point of ST]
∠ SOU = ∠ TOV  [Vertically opposite angles]
and, UO = OV  [∵ O is the mid point of UV]
So by SAS congruence criterion, we have
Δ SOU ≡ Δ TOV
SU = VT and ∠ USO = ∠ VTO  [Corresponding parts of congruent triangle are equal]
Now, SU and VT are two lines intersected by a transversal ST such that ∠USO = ∠ VTO, i.e. alternative angles are equal.
 Therefore SU is parallel to VT. 

Answer 2

in Δs SOU and TOV,   ∠UOS = ∠VOT       (vertical angles)
We know  UO  || OV   and  OS || OT.

let us see the ratio of these sides:  
  OV/OT = 1.8/6.3 = 2/7
  OU / OS = 1.2 / 4.2 = 2/7

As the ratios are equal also, it means both triangles are similar as included angle is same and the ratio of corresponding parallel sides is equal.
 
  So  SU || VT.