Geodesic
The properties of mosaic pentagons with a pair of equal adjacent edges
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 1380-1412.

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It was obtained by O.G. Bagina the complete classification of convex mosaic pentagons, admitting normal (edge to edge) tilings, in 2011–2012. The classification includes 8 types of such pentagons. In the proof of the completeness of this list the following fact was used. If a convex pentagon tiles the plane normally, belongs only the first type of the list and has only a pair of equal adjacent edges, that is, the angles and edges of this pentagon satisfy the relations $C_0 = C_1, x_2 + x_3 = 180^\circ$, then it angles satisfy also the relation $x_0 + 2x_1 = 360^\circ$. But this statement has not been proven. This paper fills this gap.
Keywords: convex pentagon, mosaic pentagon, tiling the plane, normal tiling. 
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O. G. Bagina. The properties of mosaic pentagons with a pair of equal adjacent edges. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 1380-1412. http://geodesic.mathdoc.fr/item/SEMR_2017_14_a66/

[1] O.G. Bagina, “Tiling the plane with convex pentagons”, KemSU Bulletin, 2011, no. 4(48), 63–73

[2] O.G. Bagina, “Convex pentagons which tile the plane (types: 11112, 11122)”, Siberian Electronic Mathematical Reports, 9 (2012), 478–530 | MR | Zbl

[3] D. Schattschneider, “Tiling the Plane with Congruent Pentagons”, Math. Magazine, 51 (1978), 29–44 | DOI | MR | Zbl

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