Question 1018591
Let m be the midpoint of ad and ce.
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*[illustration Parallel.png].
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Then,
{{{am=dm}}}
{{{cm=em}}}
Using the Pythagorean theorem,
{{{am^2+em^2=ae^2}}}
{{{cm^2+dm^2=cd^2}}}
Substituting,
{{{em^2+am^2=cd^2}}}
So then,
{{{ae^2=cd^2}}}
{{{ae=cd}}}
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Look at the continuation of the red and black lines to the left of point {{{a}}}.
That angle H is equal to angle G because by trigonometry,
{{{tan(H)=em/am}}}
{{{tan(G)=cm/dm}}}
and
{{{tan(G)=em/am}}}
So,
{{{tan(G)=tan(H)}}}
{{{G=H}}}
So when corresponding angles (G,H) are equal (from geometry), the lines are parallel.
So {{{ae}}} is parallel to {{{cd}}}