Question

eh bisects ∠fhj. complete the proof that △egh ≅ △eih.
1 eh bisects ∠fhj
2 ∠i ≅ ∠g
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 △egh ≅ △eih

Explanation:

Step1: Given

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

Step2: Given

Statement 2 \( \angle I\cong\angle G\) is given (marked in the diagram).

Step3: Vertical Angles Theorem

Vertical angles are congruent. So, \( \angle IHJ\cong\angle FHG\) (vertical angles).

Step4: Definition of Angle Bisector

Since \( \overleftrightarrow{EH}\) 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

From steps 3 and 5, substitute \( m\angle IHJ\) with \( m\angle FHG\) (since \( \angle IHJ\cong\angle FHG\)), so \( m\angle EHI=m\angle EHF + m\angle FHG\).

Step8: Substitution Property

From steps 6 and 7, since \( m\angle EHI=m\angle EHF + m\angle FHG\) and \( m\angle EHG=m\angle EHF + m\angle FHG\), then \( m\angle EHG=m\angle EHI\).

Step9: Reflexive Property of Congruence

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

Step10: ASA (Angle - Side - Angle) Congruence Criterion

In \( \triangle EGH\) and \( \triangle EIH\), we have \( \angle G\cong\angle I\) (statement 2), \( \overline{EH}\cong\overline{EH}\) (statement 9), and \( \angle EHG\cong\angle EHI\) (statement 8). So, by the ASA (Angle - Side - Angle) congruence criterion \( \triangle EGH\cong\triangle EIH\).

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. ASA (Angle - Side - Angle) Congruence Criterion