Question

Please help me it’s math explain

Answer 1

Since we know two angles and the side between them are congruent (look at steps 3 and 4), we can say the triangles are congruent through ASA. Thus, the gray box is "ASA Congruence."

Answer 2

The short answer, that will go in the blank space, is ASA Property of Congruence

Here's how I got that answer: 

We're at statement 5. Specifically we want the reason for why statement 5 is true. So we have to back up a few steps to look up at statement 3 and statement 4. With statement 3, we're proving that angle 2 and angle 5 are congruent, so are angles 1 and 4. So that takes care of two pairs of angles; hence the two "A" letters in ASA

Then in statement 4, we prove that BD is congruent to itself. There's not much to prove here as it is more like an axiom: any segment is congruent to itself through the reflexive property. This takes care of the "S" part in ASA

Putting that all together, we have two pairs of angles congruent with the pair of sides that are sandwiched between the angles. So that's why we use ASA