Question

The pentagon at the right is equilateral and equiangular.a. What two triangles must be congruent to prove HB¯≅HE¯?b. Write a proof to show HB¯≅HE¯.

Right triangles and trigonometry
ANSWERED
asked 2020-11-22

The pentagon at the right is equilateral and equiangular.
a. What two triangles must be congruent to prove \(\overline{HB}\cong \overline{HE}\)?
b. Write a proof to show \(\overline{HB}\cong \overline{HE}\)

Answers (1)

2020-11-23

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

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