Elementary Geometry for College Students (6th Edition)

1. Given 2. $\overline{HJ}\bot\overline{KL}$. So 2 angles created from that must be right angles. 3. Identity 4. $\triangle HJK\cong\triangle HJL$ 5. $\overline{JK}\cong\overline{JL}$
1. $\overline{HJ}\bot\overline{KL}$ and $\overline{HK}\cong\overline{HL}$ These statements are already mentioned in the exact same way in the given information of the exercise. So here, we would fill 1. Given 2. $\angle HJK$ and $\angle HJL$ are right $\angle$s From above, we notice that $\overline{HJ}\bot\overline{KL}$. Therefore, 2 angles created from that must be right angles. So here, we would fill 2. $\overline{HJ}\bot\overline{KL}$. So 2 angles created from that must be right angles. 3. $\overline{HJ}\cong\overline{HJ}$ A line is congruent with itself, by identity. So, here, we would fill 3. Identity 4. HL HL is the method to prove triangles congruent when 2 triangles are right ones. Here we see that since $\angle HJK$ and $\angle HJL$ are right $\angle$s, $\triangle HJK$ and $\triangle HJL$ are right triangles. And the leg and hypotenuse of $\triangle HJK$ are congruent with those of $\triangle HJL$, which means these two triangles are congruent. So here, we would fill $4. \triangle HJK\cong\triangle HJL$ 5. CPCTC Now, the purpose of the exercise is to prove $\overline{KJ}\cong\overline{JL}$. Since we already prove that $\triangle HJK\cong\triangle HJL$, according to CPCTC, any corresponding sides and angles of both triangles are congruent. That means $\overline{JK}\cong\overline{JL}$ So, here, we would fill 5. $\overline{JK}\cong\overline{JL}$