I can give you the answers; making you understand is a taller order.
1. Given
Proofs usually start with the givens
2. Corresponding Angle Theorem
Two sides of the triangle are transverses to parallel lines so each generates congruent angles.
3. AA (angle angle) Similarity Theorem
If a pair of triangles has two congruent pairs of angles, the third pair is congruent as well and the triangles are similar.
4. Corresponding sides of similar triangles are proportional
All the linear dimensions in a similar triangle are scaled by the same factor.
5. Segment Addition Postulate
This one is so basic sometimes it's not even mentiond. But it's really important. When we have a segment with a distinguished point dividing it into two shorter segments, the length of the whole is the sum of the lengths of the two parts.
6. That's the algebra one, (a+b)/c=a/c+b/c
It basically says we can add fractions with a common denominator by adding their numerators.
7. Subtract 1 ...
More algebra rules
8. Take reciprocals
More algebra
