Final answer:
A rectangle is the quadrilateral in which all the diagonals are always congruent, and this can be proven using the properties of right triangles created by the diagonals of the rectangle.
Step-by-step explanation:
The question 'In what quadrilateral are all the diagonals always congruent?' is asking which type of four-sided figure has diagonals that are always equal in length to each other. In geometry, a rectangle is the quadrilateral where this property holds true. Each diagonal in a rectangle is congruent, meaning they have the same length, which can be proven through the use of the Pythagorean theorem applied to the right triangles that form when a rectangle is bisected by a diagonal.
A detailed example demonstrating this would involve taking a rectangle with width 'w' and length 'l'. By drawing a diagonal we create two right-angled triangles. Since the sides of the rectangles are also the sides of these triangles, we have that these triangles are congruent due to the Side-Angle-Side (SAS) theorem. Therefore, the diagonals of the rectangle, which are the hypotenuses of these triangles, are of equal length. So, any time we have a rectangle, we can be certain that the diagonals are congruent.