Is this problem, by chance, written by someone whose native language is not English? I do not mean that as ridicule. Where I live that is more the rule than the exception. Chinese-speaking teachers use English to teach mathematics to Chinese-speaking students.

You might take this up with whoever gave you the problem. Propositions (ii) and (iv) are both untrue. They cannot be proved.

(ii)

I am taking "≡" to indicate congruence (≅).

While it is true that ∆APD ≅ ∆CPD, ∆APD certainly is not congruent to ∆PDC. It is generally accepted that corresponding vertices of congruent figures must be listed in the same order. It was not always like that, but that convention has been around for a very long time now.

(iv)

An equiangular triangle is one in which all angles are equal. It is the equivalent of an equilateral triangle. That is not possible in a triangle having a right angle. My best guess is that the word "equiangular" is being used to indicate a relation between two triangles in which corresponding angles are equal. That condition would make the triangles similar, not equiangular.

Follow-up:

I feel that I should back away from my statements on part (iv). Here it is only 13 hours later, and a student has dropped off homework using the word "equiangular" in the same way your posted problem does. I found out he got that from his workbook. I still do not like that usage, but if they are using it in textbooks now, then I suppose I should get used to it.