Rectangles: the Diagonals are of Equal Length

Rectangles are a special type of parallelogram, in which all the interior angles measure 90°. Because all rectangles are also parallelograms, all the properties of parallelograms are also true for rectangles, too:

• The opposite sides of rectangles are equal
• The diagonals of rectangles bisect each other
• Any two adjacent angles are supplementary (obviously, since they all measure 90°)
• The opposite angles are equal (again, obviously, since all interior angles measure 90°)

But because the angles are all equal, there is an additional property of rectangles that we will now prove – that the diagonals of a rectangle are equal in length.

Problem

In a rectangle ABCD, prove that the diagonals are of equal length: prove AC = DB

Strategy

Like many other geometry problems where we need to prove that two line segments are of equal lengths, we will turn to triangle congruency as our go-to tool.

Here, we don’t even need to construct any special triangles, because the diagonals themselves have defined the triangles. For example, the two triangles ΔABD and  ΔDCA, in which the diagonals form corresponding sides.

Don’t worry if you didn’t see this immediately, or if you chose other triangles – it is easy to prove this property of rectangles using other combinations of triangles, such as ΔABC and ΔDCB or ΔDAB and ΔCBA; any two pairs will do, as long as the diagonals are the corresponding sides in each triangle.

Now, having picked our triangles, we will rely on the properties of parallelograms to show triangle congruency. We can also prove this from scratch, repeating the proofs we did for parallelograms, but there’s no need. A rectangle is a parallelogram, and we can save time and effort by relying on general parallelogram properties that we have already proven.

So, looking at the triangles ΔABD and  ΔDCA, they have one common side – AD. We also know that AB= CD as they are opposite sides in a parallelogram.

And from the definition of a rectangle, we know that all the interior angles measure 90° and are thus congruent- and we can prove the triangle congruency using the Side-Angle-Side postulate.

Proof

(1) AD= AD //common side

(2) AB= CD                                 // Opposite sides in a rectangle (parallelogram).

(3) m∠BAD = m∠CDA=90°         // Definition of rectangle

(4) ∠BAD ≅ ∠CDA                      // (3) and definition of congruent angles

(5) ΔBAD ≅ ΔCDA                     // Side-Angle-Side postulate.

(6) AC = DB                                // Corresponding sides in congruent triangles (CPCTC)