Hey there!
First of all we can look at the given statement and see that the first statement in the proof has a typo, it should say line AD and not line AC.
Anyways, let's continue:
1. Given. This is the given statement we have.
2. Vertical angle conjecture. Non-adjacent angles that are formed by two intersecting lines are always congruent, or equal in measure.
3. Proportion of corresponding sides. In the diagram, BC has a length of 5 and EB has a length of 15. This can be written as a ratio as a fraction, which can be reduced to 1/3.
4. [tex]\frac{\overline{BD}}{\overline{BA}} = \frac{4}{12} = \frac{1}{3}[/tex] BD has a length of 4 and BA has a length of 12. Written as a fraction this can be reduced to 1/3.
5. Both are equal to 1/3, and 1/3 is equal to 1/3.
6. SAS/Side angle side theorem. If two corresponding sets of sides have the same proportions and the included angle is congruent, then the triangles are similar. In step 5 we prove the sides have the same proportions, and in step 2 we prove that the angle is congruent.
Thus proved.
Hope this helps!
