I have been asked to do a two column proof for a trapezoid. I am stumped on what to do. I am homeschooled and I am using a tough curriculum so the book I have for geometry assumes you know how to do everything or figure it out on your own. I would really appriciate some help please!!

figure,

question: Given: ROSE is a trapezoid with bases OS and RE.

Prove, OI • RI = EI • SI

and this is supposed to be a two column proof.

Thank you very very much!!

If a purely geometric solution is required,

you can use the following method....

Side OS is parallel to side RE.

\(\displaystyle |\angle{RIE}|=|\angle{OIS}|\)

\(\displaystyle |\angle{OSR|}=|\angle{OSI}|=|\angle{SRE}|=|\angle{IRE}|\)

\(\displaystyle |\angle{SOE}|=|\angle{SOI}|=|\angle{REO}|=|\angle{REI}|\)

Hence, triangles OSI and REI are equiangular.

Therefore triangle REI is a magnified version of triangle OSI.

Hence if we turn triangle OSI upside-down, we can compare corresponding sides, if we also turn it back to front.

Hence side OI corresponds to IE, SI corresponds to IR and SO corresponds to RE.

Each side of OSI is magnified by the same scale factor to become the corresponding side of triangle REI.

\(\displaystyle m|OI|=|IE|\ \Rightarrow\ m=\frac{|IE|}{|OI|}\)

\(\displaystyle m|SI|=|IR|\ \Rightarrow\ m=\frac{|IR|}{|SI|}\)

\(\displaystyle m|OS|=|RE|\ \Rightarrow\ m=\frac{|RE|}{|OS|}\)

\(\displaystyle \frac{|IE|}{|OI|}=\frac{|IR|}{|SI|}\ \Rightarrow\ |IE||SI|=|OI||IR|\)