Question

hk bisects ∠ihj. complete the proof that ∠j ≅ ∠i.
statement:

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. ∠j ≅ ∠i

reason: (blank for each statement)

Explanation:

Step1: Identify Given Information

Given that \( \overleftrightarrow{HK} \) bisects \( \angle IHJ \), so \( \angle IHK \cong \angle JHK \) (Definition of Angle Bisector). Also, from the diagram, \( \overline{HJ} \cong \overline{HI} \) (marked with congruency ticks), and \( \overline{HK} \cong \overline{HK} \) (Reflexive Property of Congruence).

Step2: Prove Triangle Congruence

By the Side - Angle - Side (SAS) Congruence Criterion, in \( \triangle HIK \) and \( \triangle HJK \):

• \( \overline{HJ} \cong \overline{HI} \) (Given, from the diagram's congruency marks)
• \( \angle IHK \cong \angle JHK \) (Definition of Angle Bisector, since \( HK \) bisects \( \angle IHJ \))
• \( \overline{HK} \cong \overline{HK} \) (Reflexive Property, a segment is congruent to itself)

So, \( \triangle HIK \cong \triangle HJK \) by SAS.

Step3: Corresponding Parts of Congruent Triangles

Since \( \triangle HIK \cong \triangle HJK \), their corresponding angles are congruent. The corresponding angles \( \angle J \) (in \( \triangle HJK \)) and \( \angle I \) (in \( \triangle HIK \)) are congruent. So, \( \angle J \cong \angle I \) (Corresponding Parts of Congruent Triangles are Congruent - CPCTC).

Answer:

To complete the proof:

1. Statement 1: \( \overleftrightarrow{HK} \) bisects \( \angle IHJ \)

Reason: Given

1. Statement 2: \( \overline{HJ} \cong \overline{HI} \)

Reason: Given (from the diagram's congruency marks)

1. Statement 3: \( \angle IHK \cong \angle JHK \)

Reason: Definition of Angle Bisector (since \( HK \) bisects \( \angle IHJ \))

1. Statement 4: \( \overline{HK} \cong \overline{HK} \)

Reason: Reflexive Property of Congruence

1. Statement 5: \( \triangle HIK \cong \triangle HJK \)

Reason: SAS (Side - Angle - Side) Congruence Criterion (using statements 2, 3, 4)

1. Statement 6: \( \angle J \cong \angle I \)

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