Question 1:
Since the diagonal is an angle bisector for both opposite angles and each set of opposite sides are parallel, this shape is a rhombus.
Question 2:
Since this is a rectangle, the diagonals are equal. Thus, each half of the diagonal must be equal to each other as well.
5x - 5 = 3x + 3
Subtract both sides by 3x
2x - 5 = 3
Add both sides by 5
2x = 8
Divide both sides by 2
x = 4
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