5. triangle abc is a right triangle with cd and ab angle c is a right triangle. by the △acb∼△adc and △acb∼△cdb. since similar triangles have sides, bcba=bdbc and acab=adac . using cross multiplication gives the equations (bc)2=(bd)(ba) and (ac)2=(ad)(ab). adding these together gives (bc)2+(ac)2=(bd)(ba)+(ad)(ab). factoring out the common segment gives (bc)2+(ac)2=(ab)(bd+ad). using gives (bc)2+(ac)2=(ab)(ab), which simplifies to (bc)2+(ac)2=(ab)2 . first paranthese sas or aas or sss second paranthese congruent or proportional third paranthese sa or cpctc

For the first parenthesis, it would be the Angle Angle Similarity Postulate since we have essentially pairs of corresponding congruent angles that match up, allowing for these three similar triangles

The second parenthesis would be "proportional". Any time you have similar triangles, the sides form a ratio. For example if ABC is similar to DEF, then AB/DE = BC/EF

Finally the third parenthesis would be "Segment Addition Postulate" possibly shorted and abbreviated to simply SA. This idea is that we can glue two lines together to form a larger line. In this case, AD+DB turns into AB.