Question

hk bisects ∠ihj and ( overline{hj} cong overline{hi} ). complete the proof that ( overline{ik} cong overline{jk} ).
(there is a triangle diagram with vertices h, i, j and a point k on ij, and a table with statements: 1. hk bisects ∠ihj, 2. ( overline{hj} cong overline{hi} ), 3. ∠ihk ≅ ∠jhk, 4. ( overline{hk} cong overline{hk} ), 5. ( \triangle hik cong \triangle hjk ), 6. ( overline{ik} cong overline{jk} ), and empty reason columns.)

Explanation:

Step1: Given

$\overleftrightarrow{HK}$ bisects $\angle IHJ$, so $\angle IHK=\angle JHK$. Also, $\overline{HJ}\cong\overline{HI}$ (given).

Step2: Reflexive Property

$\overline{HK}\cong\overline{HK}$ (a segment is congruent to itself).

Step3: SAS (Side - Angle - Side) Congruence

In $\triangle HIK$ and $\triangle HJK$, we have $\overline{HI}\cong\overline{HJ}$ (given), $\angle IHK\cong\angle JHK$ (angle - bisector property), and $\overline{HK}\cong\overline{HK}$ (reflexive property). By the SAS (Side - Angle - Side) congruence criterion, $\triangle HIK\cong\triangle HJK$.

Step4: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since $\triangle HIK\cong\triangle HJK$, then $\overline{IK}\cong\overline{JK}$ (by CPCTC).

Answer:

1. Given; 2. Given; 3. Definition of angle bisector; 4. Reflexive property of congruence; 5. SAS (Side - Angle - Side) congruence; 6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)