Question

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

Explanation:

Step1: Identify corresponding - angles

Since $\overleftrightarrow{FG}\parallel\overleftrightarrow{EH}$, $\angle HEI\cong\angle FGI$ because corresponding angles formed by parallel lines are congruent.

Step2: Use given congruence

We are given that $\angle HEI\cong\angle F$. Also, if two angles are congruent to the same angle, they are congruent to each other. So, since $\angle HEI\cong\angle F$ and $\angle HEI\cong\angle H$ (from the property of angles in the triangle with $\angle HEI$ and $\angle H$), then $\angle F\cong\angle H$ by the transitive property of congruence.

Step3: Substitute congruent - angles

Since $\angle HEI\cong\angle F$ and $\angle F\cong\angle H$, by the transitive property of congruence, $\angle HEI\cong\angle H$.

Step4: Prove the final congruence

We know $\angle HEI\cong\angle FGI$ and $\angle HEI\cong\angle H$. By the transitive property of congruence, $\angle H\cong\angle FGI$.

Answer:

1. Corresponding angles of parallel lines are congruent; 4. Transitive property of congruence; 5. Transitive property of congruence