# How to show these lines are parallel?

We know that: $$\angle ABO = \angle OBC$$ $$\angle DCO = \angle OCB$$ Prove: $$AB\parallel CD$$

I have tried so many things: First I continued lines $BO$ and $CO$ to reach $CD$ and $AB$. Then I tried adding a parallel line to $AB$ that goes through point $O$. Also I tried adding a median to $BC$ that goes through point $O$, and in another attempt I tried it with an angle bisector. Also I have tried continuing lines $AB$ and $CD$ the other way and adding parallel lines to $BO$ and $OC$ which goes through points $C$ and $B$.

First I thought it seems so natural as just part of a Rhombus, but I found out that it isn't that easy as I thought it would be.

• Imagine moving $O$ up the page, so that $\alpha, \beta$ get close to 90°. Then the two side lines will be nowhere near parallel. – Joffan Feb 3 '17 at 18:49
$$O_1=180-(\alpha+\beta)\\ \to o_2=180-o_1=\alpha+\beta\\$$now in $\triangle ODC$ we have $$o_2+D+c=180 \\D=180-(\alpha+\beta)-\beta$$ Note that $AB||CD$ only when $\hat{D}=\alpha$ so $$D=180-(\alpha+\beta)-\beta=\alpha \to 2(\alpha+\beta)=180 \\\to \alpha+\beta=90$$ This condition is necessary for $AB||CD$
It looks like fake. Think about a regular hexagon with vertex $A,B,C,D$ (and others of course) and center $O$. Then all the angles are $60°$ but $AB,CD$ are not opposite edges, so no parallel