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¯.

Question
The pentagon at the right is equilateral and equiangular.
a. What two triangles must be congruent to prove \(\displaystyle{H}{B}¯≅{H}{E}¯\)?
b. Write a proof to show \(\displaystyle{H}{B}¯≅{H}{E}¯.\)

Answers (1)

2020-11-23
a. HB¯ and HE¯ are side lengths of △HBC and △HDE. Therefore, to prove HB¯≅HE¯, we must prove \(\displaystyle△{H}{B}{C}≅△{H}{D}{E}.\)
b. Proof Outline: Since the pentagon is equilateral, then we know ED¯≅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.\(\displaystyle{<}{B}{C}{D}≅{<}{E}{D}{C}\)</span> 4. Def. of equiangular
5.\(\displaystyle{C}{D}≅{C}{D}\) 5. Reflexive Property
6.\(\displaystyle△{B}{C}{D}≅{E}{D}{C}\) 6. SAS
7.\(\displaystyle{<}{H}{B}{C}≅{<}{H}{E}{D}\)</span> 7. CPCTC
8.\(\displaystyle△{H}{B}{C}≅△{H}{D}{E}\) 8.AAS
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