Check the picture attached.
Let m(BAE)=m(ACD)=α
(BAE and ACD are congruent, since they are alternate interior angles, or Z angles)
Let m(ABE)=β.
So in triangle ABE, the measures of the angles are 90, α and β degrees.
This means that m(BCE)=β, since the 2 other angles of triangle BCE are 90 and α degrees.
thus, we have the similarity of triangles ABE and BCE,
so the following rations are equal:
so
so
(inches)
Remark, we can also apply Euclid's theorem directly.
Let m(BAE)=m(ACD)=α
(BAE and ACD are congruent, since they are alternate interior angles, or Z angles)
Let m(ABE)=β.
So in triangle ABE, the measures of the angles are 90, α and β degrees.
This means that m(BCE)=β, since the 2 other angles of triangle BCE are 90 and α degrees.
thus, we have the similarity of triangles ABE and BCE,
so the following rations are equal:
so
so
Remark, we can also apply Euclid's theorem directly.