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

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 $$\overline{HB}\cong \overline{HE}$$

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