Since segments ST and VW are parallel, triangles VPW and SPT are similar. This is due to AAA (angle-angle-angle) theorem. In order to see this, we can draw the following picture
where we can see that angle V and angle S are the same, angle W and angle T are the same and angle P is the same in both triangles.
Now, since the triangles are similar, the following ratio must be preserved:
[tex]\frac{PS}{10}=\frac{PS+6}{17.5}[/tex]
If we move 10 to the right hand side and 17.5 to the left hand side, we get
[tex]17.5\cdot PS=10\cdot(PS+6)[/tex]
which is equal to
[tex]17.5PS=10PS+60[/tex]
If we move 10PS to the left hand side as -10PS, we obtain
[tex]\begin{gathered} 17.5PS-10PS=60 \\ 7.5PS=60 \\ PS=\frac{60}{7.5} \\ PS=8 \end{gathered}[/tex]
Now, since PT has the same length as PS and TW has the same length as SV, the answer is
[tex]\begin{gathered} PS=PT=8 \\ SV=TW=6 \end{gathered}[/tex]
