# A1....A9 be a nine-sided regular polygon with side length 2. The difference between the lengths of A1A5 and A2A4 equals 2

Let A1A2A3 ....A9 be a nine-sided regular polygon with side length 2 units. The difference between the lengths of the diagonals A1A5 and A2A4 equals $$2$$.

I can prove it very easily if I could have proved that the side $$A_2 A_4$$ is parallel to $$A_1 A_5$$ in the following picture.

Is it possible to prove?

• The figure you drew has ten sides. Nov 20 '19 at 12:05

Yes. $$\angle415\cong\angle142$$, since they are inscribed angles that intercept congruent arcs of $$\frac{360^\circ}9$$. Since those angles are alternate interior angles of the lines $$\overline{24}$$ and $$\overline{15}$$ cut by transversal $$\overline{14}$$, we conclude that $$\overline{24}\parallel\overline{15}$$.