Does this theorem about two lines and two transversals have a name?

Question

I've got this theorem right here. It involves two arbitrary lines $AB$ and $CD$ and two transversals $AC$ and $BD$, intersecting in between these lines.

The theorem (or lemma, I don't know) in question states that $\alpha+\beta = \varphi + \theta$.

The simplest proof of which I know is based on the Sum of Angles of Triangle equals Two Right Angles: two transversals intersecting in between two lines form two triangles with these lines. Then sums of angles of these triangles are equal to one another. One pair of angles is equal as vertical angles, so the sum of two remaining angles in one triangle should be equal to the sum of two remaining angles, which was to be proven.

I've got three questions in regards to this fact:

• Does this fact (is it a lemma or a theorem?) has a name? I thought of something related to bowtie, butterfly, or hourglass, but there are numerous facts names after these objects.
• What is the correct way to formulate this fact by using parallel lines and transversals?
• If there's any proof of it only through parallel lines and transversals, then what is that proof?

Answer 1

Constant vertex rotation angle theorem, if you will:

$$ \gamma= \alpha-\theta= \varphi - \beta \to \ \alpha+\beta= \varphi+\theta $$

Arbitrary lines AB,CD with fixed vertex V rotating arbitrarily, variable transversals AC,BD.

Answer 2

Just have a look at this drawing, where the left green angle equals the right green angle (quite obvious):

The sum of the angles of the triangles $AEC$ and $DEB$ are equal to $180°$.

Hence, the sum of the angles $A+C$ equals $B+D$.

Why would something, so obvious, need to be a special theorem?