a. \(\overline{HB}\) and \(\overline{HE}\) are side lengths of \(\triangle HBC\ and\ \triangle HDE\). Therefore, to prove \(\overline{HB}\cong \overline{HE}\), we must prove \(\displaystyle△{H}{B}{C}≅△{H}{D}{E}.\)

b. Proof Outline: Since the pentagon is equilateral, then we know \(\overline{ED}\cong \overline{BC}\). Vertical angles are congruent so we also know \(\displaystyle∠{E}{H}{D}≅∠{C}{H}{B}\). This is not enough to prove the triangles are congruent though. Since the pentagon is equiangular, we know \(\displaystyle∠{B}{C}{D}≅∠{E}{D}{C}\). We can then prove that \(\displaystyle△{B}{C}{D}≅△{E}{D}{C}\) by SAS. Using CPCTC, we then have \(\displaystyle∠{H}{B}{C}≅∠{H}{E}{D}\). We now have two pairs of congruent corresponding angles and a pair of congruent nonincluded sides so then \(\displaystyle△{H}{B}{C}≅△{H}{D}{E}\) by AAS.

Proof:

Statements Reasons

1.ABCDE is an equilateral and 1. Given equiangular pentagon

2.\(\displaystyle{E}{D}≅{B}{C}\) 2. Def. of equilateral

3.\(\displaystyle∠{E}{H}{D}≅∠{C}{H}{B}\) 3. Vertical Angles Theorem

4.\(\displaystyle{<}{B}{C}{D}≅{<}{E}{D}{C}\) 4. Def. of equiangular

5.\(\displaystyle{C}{D}≅{C}{D}\) 5. Reflexive Property

6.\(\displaystyle△{B}{C}{D}≅{E}{D}{C}\) 6. SAS

7.\(\displaystyle{<}{H}{B}{C}≅{<}{H}{E}{D}\) 7. CPCTC

8.\(\displaystyle△{H}{B}{C}≅△{H}{D}{E}\) 8.AAS