# 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¯.

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

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falhiblesw

a. $\stackrel{―}{HB}$ and $\stackrel{―}{HE}$ are side lengths of . Therefore, to prove $\stackrel{―}{HB}\cong \stackrel{―}{HE}$, we must prove $△HBC\cong △HDE.$
b. Proof Outline: Since the pentagon is equilateral, then we know $\stackrel{―}{ED}\cong \stackrel{―}{BC}$. Vertical angles are congruent so we also know $\angle EHD\cong \angle CHB$. This is not enough to prove the triangles are congruent though. Since the pentagon is equiangular, we know $\angle BCD\cong \angle EDC$. We can then prove that $△BCD\cong △EDC$ by SAS. Using CPCTC, we then have $\angle HBC\cong \angle HED$. We now have two pairs of congruent corresponding angles and a pair of congruent nonincluded sides so then $△HBC\cong △HDE$ by AAS.
Proof:
Statements Reasons
1.ABCDE is an equilateral and 1. Given equiangular pentagon
2.$ED\cong BC$ 2. Def. of equilateral
3.$\angle EHD\cong \angle CHB$ 3. Vertical Angles Theorem
4.$ 4. Def. of equiangular
5.$CD\cong CD$ 5. Reflexive Property
6.$△BCD\cong EDC$ 6. SAS
7.$ 7. CPCTC
8.$△HBC\cong △HDE$ 8.AAS

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