I am sorry to tell you but who ever set this question had no idea of the terms involed.View attachment 27537NAME THE ANGLES:
alternate interior to <1
corresponding to < 3
supplementary to < 1
supplementary to < 5
interior on the same side of the transversal to < 3
I think it is just a badly drawn diagram, possibly redrawn by the poster.I am sorry to tell you but who ever set this question had no idea of the terms involed.
Please look at this link. If you do you will see the there is no actual transversal in the diagram you were given.
On that page it is written "In geometry, a transversal is a line that passes through two lines in the same plane at two distinct points. Transversals play a role in establishing whether two or more other lines in the Euclidean plane are parallel. The intersections of a transversal with two lines create various types of pairs of angles: consecutive interior angles, consecutive exterior angles, corresponding angles, and alternate angles. "
Please read the whole link then post your questions.
Note that the two lines are parallel. Thus your \(Z\) only holds for parallel lines and alternate interior angles.1. Alternate angles form a Z shape.
But \(\alpha~\&~\alpha_1\) are corresponding angles as are \(\beta~\&~\beta_1\).2. Corresponding angles form an F shape (or a backwards F shape) with both under the arms of the F.
Here is the F for the alphas:Having taught axiomatic geometry for graduates, I am very sensitive to vocabulary. Wylie in his text The Foundations of Geometry uses this definition: A transversal of two coplanar lines is a line which intersect the union of the lines in two points.
Note that the two lines are parallel. Thus your \(Z\) only holds for parallel lines and alternate interior angles.
There are alternate exterior angles.
View attachment 27546
In the diagram above \(\alpha_1~\&~\gamma\) are alternate exterior angles while \(\delta_1~\&~\beta\) are alternate interior angles.
But \(\alpha~\&~\alpha_1\) are corresponding angles as are \(\beta~\&~\beta_1\).
We are hard pressed to find an \(F\) in that.