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## Answers

As all the diagonals are equal AC = BD = CE = AD = BE

Solution:

we know that ∠A = ∠B = ∠C = ∠D = ∠E = 108 degree

Let us consider ∆ABC and ∆ADE

Here,AB = AE

BC = DE

∠ABC = ∠AED = 108 degree

so, ∆ABC = ∆ADE

and AC = AD -----(1)

Similarly ∆BCD = ∆CDE

BD = CE -------- (2)

∆ ABC = ∆BCD

AC = BD -------- (3)

∆ABC = ∆ABE

BE = AC -------- (4)

Equation the equations (1), (2), (3) and (4)

AC = BD = AD = CE = BE

Thus all the diagonals of a regular polygon are equal

To prove: All the diagonals of the pentagon are equal………

i.e. AC = AD = BD = CE = BE……………

Proof:.…..

ABCDE is a regular pentagon………

Þ AB = BC = CD = DE = AE and ∠A = ∠B = ∠C = ∠D = ∠E = 108°.………

Consider ∆ABC and ∆ADE……………

AB = AE ...... (Given)……..

BC = DE ….......... (Given)..........

∠ABC = ∠AED = 108° .............. (Given)............

∴ ∆ABC ≅ ∆ADE ..........

(SAS congruency rule)..........

∴ AC = AD .... (corresponding sides of congruent triangles) -------- (1).....

Similarly ∆BCD ≅ ∆CDE.......

⇒ BD = CE -------- (2)......

∆ ABC ≅ ∆BCD ..........

⇒ AC = BD -------- (3)...........

∆ABC ≅ ∆ABE............

⇒ BE = AC -------- (4)...........

From equations (1), (2), (3) and (4), we get....

AC = BD = AD = CE = BE...........

Therefore, all the diagonals of a pentagon are equal.…………