Can the triangle inequlity extened to show the distance inequlity of a trapezium

Question

$AB // CD$. What are the angle conditions (acute, obtuse or right angle) of $a,b,c,d$ to be satisfied the inequality $ |AB+BC| > |CD|$?

$AB,BC,CD$ are distances.

Answer 1

BI think the inequality holds iff $a+\frac{b}{2}<180$. Let $P$ be a point on $AB$ where $B$ is between $P$ and $A$ such that $PB=BC$. Also Let $Q$ be the fourth vertex of parallelogram $APCQ$ that lies on $CD$ (or its extension). The desired inequality holds only when $D$ is between $C$ and $Q$ which translates to the angle inequality I mentioned.