Is My Proof Correct? Geometry?

Your proof is NOT correct: That ray BD bisects angle B does not imply that D is the midpoint of the segment AC. So, lines three through six of your proof are all incorrect. That point D is the midpoint is a consequence of what you intend to prove! Here is the correct proof:

(1) Given

(2) Angle ADB is congruent to Angle CBD (Definition of Angle Bisector)

(3) AB = CB (Definition of Isoceles Triangle) [and here I assume that AC is the base although you have not stated this]

(4) BD = BD (Reflexivity)

(5) triangle ABD is congruent to triangle CBD (SAS Postulate)

This completes the proof. But, to continue, by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), you can now say that AD = CD and angle BDA is congruent to angle BDC. As a result, D is the midpoint of AC by definition and that BD is perpendicular to AC since the angles BDA and BDC are congruent and supplementary (since they lie on the same line) and therefore must both measure 90 degrees.

you've it actually right. #4 (of the second one 1/2-- your 1/2) reads "if 2 right angles......." even as it would want to study "if 2 right triangles....." 2 triangles are similar even as each and every triangle has an perspective it truly is congruent to an perspective with the different. in the experience that they are right triangles, they each and every have one ninety degree perspective. If the each and every have one acute perspective equivalent, then the triangles are similar. as far as #5 is going, the very shown actuality that the perimeters are proportional is likewise the definition of triangles that are similar. all of it looks sturdy.