Answer to Question #413 in Geometry for Lenny

Question

The diagonals of a rectangle must be congruent. Is it true?

Answer 1

Yes, it's true. You can prove this quickly by using the side-angle-side congruency theorem.

Given rectangle ABCD, draw diagonals AC and BD. Side AD is congruent to side BC because they are opposite sides of a rectangle. Angle D is congruent to angle C since they are both right angles. And side DC is congruent to CD because it is the same segment. Using the SAS theorem, this proves triangle ADC is congruent to triangle BCD. Thus AC is congruent to BD because they are the hypotenuses of the triangles.