Will a pentagon and square tessellate?

Not a regular pentagon (equal sides) and square. But suppose the pentagons are shaped like a child's drawing of a house: a square with a triangle on top where the triangle is a right isosceles triangle (90, 45, 45 deg).

Align a line of these pentagons with their "rooves" facing up. Next put a second row of inverted pentagons on top - apex-to-apex. The holes left will be squares. So tessellate away.