Question

hi || fg and fg ≅ hi. complete the proof that δfhi ≅ δhfg. image of a quadrilateral (trapezoid) with vertices g, h, i, f and a diagonal fh. table with statements: 1. hi || fg (reason: given), 2. fg ≅ hi (reason: given), 3. ∠fhi ≅ ∠gfh, 4. fh ≅ fh, 5. δfhi ≅ δhfg; reasons for 3,4,5 are blank

Explanation:

Step1: Determine the reason for \( \overline{FH}\cong\overline{FH} \)

By the Reflexive Property of Congruence, any segment is congruent to itself. So, \( \overline{FH}\cong\overline{FH} \) because of the Reflexive Property of Congruence.

Step2: Determine the reason for \( \angle FHI\cong\angle GFH \)

Since \( \overline{HI}\parallel\overline{FG} \), by the Alternate - Interior Angles Theorem, \( \angle FHI\cong\angle GFH \).

Step3: Determine the reason for \( \triangle FHI\cong\triangle HFG \)

We have \( \overline{FG}\cong\overline{HI} \) (given), \( \angle FHI\cong\angle GFH \) (Alternate - Interior Angles Theorem), and \( \overline{FH}\cong\overline{FH} \) (Reflexive Property of Congruence). By the Side - Angle - Side (SAS) Congruence Postulate, \( \triangle FHI\cong\triangle HFG \).

Answer:

1. Given
2. Given
3. Alternate - Interior Angles Theorem
4. Reflexive Property of Congruence
5. Side - Angle - Side (SAS) Congruence Postulate