Question

Are the diagonals of a rectangle always, sometimes or never congruent?

Answer 1

Hint: Here, we need to prove that the diagonals of a rectangle are congruent or not. We will use the properties of a rectangle to prove whether the two triangles (with one diagonal as a side) are congruent or not.

Complete step by step answer:
First, we will draw a rectangle with its diagonals.

Here, ABCD is the rectangle where AC and BD are the diagonals. M is the intersection point of the diagonals AC and BD. We need to prove that AC=DB. We will use the properties of a rectangle and congruence of triangles to prove that the diagonals of a rectangle are congruent.Now, we know that the opposite sides of a rectangle are always equal.Therefore, we get
$$AB=CDandBC=AD$$
We know that all the interior angles of a rectangle are right angles.
Therefore, we get
$$\mathrm{\angle }ABC=\mathrm{\angle }BCD=\mathrm{\angle }CDA=\mathrm{\angle }DAB={90}^{\circ }$$

Now, we will prove that the triangles ABC and DCB are congruent.
In triangles, ABC and DCB, we have
$AB=CD$……………(Opposite sides of a rectangle)
$$\mathrm{\angle }ABC=\mathrm{\angle }DCB={90}^{\circ }$$............(Interior angles of a rectangle)
$BC=CB$............... (Common side)
Therefore, by S.A.S. congruence criterion, the triangles ABC and DCB are congruent.Now, the congruent parts of two congruent triangles are equal and congruent.Therefore, since the triangles ABC and DCB are congruent, we get
$$\therefore AC=DB$$

Therefore, we have proved that the diagonals of the rectangle are equal.

Note: We have used the S.A.S. congruence criterion to prove that the triangles ABC and DCB are congruent. According to the S.A.S. congruence criterion, if two corresponding sides of two triangles are equal, and the corresponding angle formed by those sides is also equal, then the two triangles are congruent.