# Find my error: Opposite angles of a trapezoid are congruent

Point out where I am wrong here cause after applying some similarity concept to a trapezoid I get a result in which the angles opposite in a trapezoid are congruent which only applies to parallelogram, I think.

1. Consider trapezoid ABCD, with parallel sides AB and CD, and its diagonals intersecting at O.
2. Alphas and betas are congruent as they result from parallel sides. Angles about O are congruent as they are vertical angles.
3. By AAA similarity test, $\triangle AOB$ and $\triangle COD$ are similar triangles.
4. By similarity, $|\overline{OA}|:|\overline{OD}|=|\overline{OB}|:|\overline{OC}|$ (proportional).
5. $\triangle AOC$ and $\triangle DOB$ are similar(?), since we have congruent angles $\angle AOC \cong \angle BOD$ with corresponding proportional sides: SAS similarity test.
6. By previous similarity, $\angle OAC \cong \angle ODB$.
7. $\angle OAC + \angle OAB = \angle ODB + \angle ODC$: the angles opposite in the trapezoid are equal.

Which part of my reasoning is wrong?

• " Triangle AOC and BOD is similar(?) since we have congruent angles AOC and BOD with corresponding proportional sides: " those sides aren't proportional tho corresponding sides of new triangles AOC and BOD. Let AO =kOD, and OB=kOC. Then AO = kAO but OC = $\frac 1k$OB $\ne$k OB \$. So AOD is not proportional to DOB. – fleablood Jun 22 '17 at 5:44
• FYI: In additional to the mistake pointed out, the mentioned trapezoid ABCD was wrongly drawn as trapezoid ABDC. – Mick Jun 22 '17 at 5:45
• I find the green markings mysterious. Usually, when two line segments are both marked with a double line across the middle, it means they are congruent. Here, the segments generally are not congruent. Perhaps that is what led you into your error. – David K Jun 22 '17 at 5:50
• Or put another way AO:OD = BO:OC. So AO:OC = BO:OD. But to have the triangles being similar we need AO:OC=DO:OB. And we do not have that. – fleablood Jun 22 '17 at 5:50
• Try or imagine drawing a trapezoid so that ABO is 100 times larger than DCO. Then AO is 100 times larger than OD but OC is 100 times SMALLER than OB. So that simply won't work. – fleablood Jun 22 '17 at 5:56