Answer
1. $HJ$ bisects $\angle KHL$
2. The bisector of an angle divides it into 2 congruent angles.
3. Given
4. A perpendicular line creates 2 corresponding right angles, which make them congruent with each other.
5. $\overline{HJ}\cong\overline{HJ}$
6. $\triangle HJK\cong\triangle HJL$
7. CPCTC.
Work Step by Step
1. Given
Since the reason states Given, the statement must be something already mentioned in the given information part.
Now, take a look at 2. $\angle JHK\cong\angle JHL$, which must be the result of the statement in 1. These congruent angles give us an intuition that $HJ$ bisects $\angle KHL$, which is exactly what is mentioned in the given information part.
So, $HJ$ bisects $\angle KHL$ is what to be filled in 1.
2. $\angle JHK\cong\angle JHL$
As mentioned above, this is the result of the statement in 1. $HJ$ bisects $\angle KHL$, which makes two resulting angles congruent with each other.
Therefore, we would fill here 2. The bisector of an angle divides it into 2 congruent angles.
3. $\overline{HJ}\bot\overline{KL}$
This statement is already mentioned in the given information part.
So, we would fill here 3. Given.
4. $\angle HJK\cong\angle HJL$
Since $\overline{HJ}\bot\overline{KL}$, two resulting angles $\angle HJK$ and $\angle HJL$ must be right angles. And since they are two corresponding right angles, they must be congruent with each other.
Here we would fill 4. A perpendicular line creates 2 corresponding right angles, which make them congruent with each other.
5. Identity
The reason mentioned Identity. So we would look a line that 2 triangles share. Here only line $\overline{HJ}$ are shared by both $\triangle HJK$ and $\triangle HJL$. So $\overline{HJ}$ is the only choice to have here.
Therefore, we would fill 5. $\overline{HJ}\cong\overline{HJ}$
6. ASA
Throughout the whole process above, we've got 2 angles and the included side of $\triangle HJK$ are congruent with 2 corresponding angles and the included side of $\triangle HJL$, which is also method ASA.
So now, we can conclude $\triangle HJK\cong\triangle HJL$ by method ASA to fill in 6.
7. $\angle K\cong\angle L$
We conclude above that $\triangle HJK\cong\triangle HJL$. $\angle K$ and $\angle L$ are 2 corresponding angles of $\triangle HJK$ and $\triangle HJL$ respectively.
Therefore, to show $\angle K\cong\angle L$, we would use method CPCTC.
So, to fill here 7. CPCTC.
