Question

∠hei ≅ ∠f and (overrightarrow{fg}paralleloverrightarrow{eh}). complete the proof that ∠h ≅ ∠fgi.
statement reason
1 ∠hei ≅ ∠f given
2 (overrightarrow{fg}paralleloverrightarrow{eh}) given
3 ∠hei ≅ ∠fgi
4 ∠f ≅ ∠h
5 ∠hei ≅ ∠h
6 ∠h ≅ ∠fgi transitive property of congruence

Explanation:

Step1: Identify corresponding - angles

Since $\overrightarrow{FG}\parallel\overrightarrow{EH}$, $\angle HEI\cong\angle FGI$ by the Corresponding Angles Postulate.

Step2: Use vertical - angles property

$\angle F\cong\angle H$ because they are vertical angles (assuming some unstated vertical - angle relationship in the figure, perhaps $\angle F$ and $\angle H$ are formed by intersecting lines in a way that makes them vertical angles).

Step3: Apply transitive property

We know $\angle HEI\cong\angle F$ (given) and $\angle F\cong\angle H$, so $\angle HEI\cong\angle H$ by the Transitive Property of Congruence.

Step4: Final transitive step

Since $\angle HEI\cong\angle FGI$ and $\angle HEI\cong\angle H$, then $\angle H\cong\angle FGI$ by the Transitive Property of Congruence.

Answer:

1. Corresponding Angles Postulate; 4. Vertical Angles are congruent; 5. Transitive Property of Congruence