Billy
Billy kisses his father at the end of the play after confessing his love. This is a metaphor for how the choices that a parent makes can play out into the belief system of a child, causing that child to think in a manner that is not the norm.
Architect
Martin being an architect is a metaphor for the fact that he makes a living designing. Thus, his being involved with an animal sexually shows that he is going against human design, against his nature.
Secrets Preferred
Ross wants to know who Martin is seeing and finds out it's a goat. When he discovers this, we understand that Ross would rather prefer not to know. And the fact that Ross represents the media in the play is a metaphor for the fact that media prefers people without their issues (to be known). So, as long as you keep them a secret you can go on air.
Paradox
Billy kisses his father, to which Martin erupts on him. This is a metaphor for the inability for one person to live with another who they believe is doing wrong, when they themselves are living in an unhealthy way. This relates to Martin's bestiality, yet his judgement is applied to Billy's homosexuality.
Tragedy
The killing of the goat by Stevie at the end of the play is a metaphor for tragedy. She invokes the great suffering of her husband at the loss of his love, and she inflicts this pain, the same pain she feels by his betrayal of their marriage.
