Prove a cyclic quad where there are parallel lines

Question

In the following diagram, PT and PU are tangents. Prove that MUPT is a cyclic quadrilateral.

In order to use the $\text{(ext $\angle = $ int opp $\angle$)}$ rule:

$\widehat{U_4} = \widehat{T_2}\quad\text{(tan chord)}$

but now I can't prove that $\widehat{T_1} = \widehat{U_3}$.

I have drawn this in GeoGebra, so I can graphically see that it is true.

Answer 1

A more complete version of qsmy's answer is this*:

Let the intersection of the rays extending from $PU$ and $TW$ be $Q$.

$\angle UPM=\angle QUW\quad\text{(corr. $\angle$'s, $PM||UW$)}$

$\angle QUW=\angle UTW\quad\text{(tan chord)}$

$\therefore \angle UPM = \angle UTW$

$\therefore MUPT \text{ is a cyclic quad}\quad\text{(line segment subtends $= \angle$'s)}$

To make the last line clear, the lemma used is this: "If a line segment subtends equal angles at two other points on the same side of the line segment, then these four points are concyclic."

*qsmy and others did not want that answer to be edited.

Answer 2

Let the ray where $PU$ is be $PQ$. It is actually simply $∠UPM=∠QUW=∠UTW$.