Question

eh bisects ∠fhj. complete the proof that △ehi ≅ △ehg.
1 eh bisects ∠fhj
2 ∠g ≅ ∠i
3 ∠ihj ≅ ∠fhg
4 ∠ehj ≅ ∠ehf
5 m∠ehi = m∠ehj + m∠ihj
6 m∠ehg = m∠ehf + m∠fhg
7 m∠ehi = m∠ehf + m∠fhg
8 m∠ehg = m∠ehi
9 eh ≅ eh
10 △ehi ≅ △ehg

Explanation:

Step1: Given

Statement 1 is given as \( \overrightarrow{EH} \) bisects \( \angle FHJ \).

Step2: Given

Statement 2 \( \angle G\cong\angle I \) is given.

Step3: Vertical angles congruence

Vertical angles \( \angle IHJ\) and \( \angle FHG \) are congruent (Vertical Angles Theorem).

Step4: Definition of angle - bisector

Since \( \overrightarrow{EH} \) bisects \( \angle FHJ\), by the definition of an angle - bisector \( \angle EHJ\cong\angle EHF\).

Step5: Angle - addition postulate

For \( \angle EHI\), by the angle - addition postulate \( m\angle EHI=m\angle EHJ + m\angle IHJ\).

Step6: Angle - addition postulate

For \( \angle EHG\), by the angle - addition postulate \( m\angle EHG=m\angle EHF + m\angle FHG\).

Step7: Substitution property

Substitute \( m\angle EHJ\) with \( m\angle EHF\) and \( m\angle IHJ\) with \( m\angle FHG\) (from steps 3 and 4) into the equation for \( m\angle EHI\). So \( m\angle EHI=m\angle EHF + m\angle FHG\).

Step8: Substitution property

Since \( m\angle EHI=m\angle EHF + m\angle FHG\) and \( m\angle EHG=m\angle EHF + m\angle FHG\) (from steps 6 and 7), by substitution \( m\angle EHG = m\angle EHI\), which means \( \angle EHG\cong\angle EHI\).

Step9: Reflexive property

For segment \( \overline{EH}\), by the reflexive property of congruence \( \overline{EH}\cong\overline{EH}\).

Step10: AAS (Angle - Angle - Side) congruence criterion

In \( \triangle EHI\) and \( \triangle EHG\), we have \( \angle I\cong\angle G\) (step 2), \( \angle EHI\cong\angle EHG\) (step 8), and \( \overline{EH}\cong\overline{EH}\) (step 9). By the AAS (Angle - Angle - Side) congruence criterion, \( \triangle EHI\cong\triangle EHG\).

Answer:

1. Given
2. Given
3. Vertical Angles Theorem
4. Definition of angle - bisector
5. Angle - addition postulate
6. Angle - addition postulate
7. Substitution property
8. Substitution property
9. Reflexive property of congruence
10. AAS (Angle - Angle - Side) congruence criterion