The problem asks for the appropriate rephrased statement to prove that the diagonals in a rectangle are congruent. Let's analyze the choices:

1. Choice A: This states that if the opposite sides of a quadrilateral are parallel, then all angles are $90^\circ$. This describes a property of rectangles but doesn't directly address the congruence of the diagonals.

2. Choice B: This says that if all angles in a quadrilateral are $90^\circ$, then opposite sides are parallel, which essentially defines a rectangle. However, this still doesn't directly prove that the diagonals are congruent.

3. Choice C: This claims that if the diagonals are congruent, then all angles are $90^\circ$. This statement is misleading because the congruence of diagonals doesn't guarantee that the quadrilateral is a rectangle (e.g., an isosceles trapezoid also has congruent diagonals but is not a rectangle).

4. Choice D: This states that if all angles are $90^\circ$ (which makes the quadrilateral a rectangle), then the diagonals are congruent. This is the correct approach to proving that in a rectangle, the diagonals are congruent.

Correct Answer:

Choice D is the correct rephrased statement for Branson's proof: "In quadrilateral $ABCD$, if $m\angle A = m\angle B = m\angle C = m\angle D = 90^\circ$, then $\overline{AC} \cong \overline{BD}$."

Would you like further clarification or have any questions? Here are some related questions you might find useful:

1. How would you prove that a quadrilateral with congruent diagonals and perpendicular bisectors is a rectangle?
2. What other properties of diagonals are specific to rectangles?
3. How can you use vector geometry to prove the congruence of diagonals in a rectangle?
4. How does the congruence of diagonals relate to other types of quadrilaterals like rhombuses and squares?
5. What is the role of symmetry in proving properties of rectangles?

Tip: When proving properties of geometric shapes, it’s essential to clearly link the defining characteristics of the shape to the property you're trying to prove!