Yes, that's right. The corresponding angles will be diagonally opposite each other.
Parallelogram
It is a square
It's a rhombus
Any parallelogram. That includes any rhombus, rectangle, and square. It could also go on to include any regular polygon with an even number of sides. Those all have more than two pairs of parallel sides and pairs of equal angels.
An isosceles triangle. It has two pairs of equal sides, which are also not parallel.
A kite has two pairs of equal sides with no parallel sides.
A sqare two pairs of parallel sides each corner is a right angle and all sides are equal.
A regular square has 2 pairs of equal opposite parallel sides
If you are not a square then you are a rectangle.
Square
rhombus
rectangle