Question

Prove that if the diagonals of a parallelogram are congruent, then it must be a rectangle. (Hint: you might need to use the fact that equilateral angles add up to 360°, or it’s also possible to do it with triangle sum or EAT. Both of these methods are recommended.)

Answer 1

We will have the following:

We will have the paralelogram ABCD and it has the diagonals AC congruent with BD.

Now, we will have that:

*Segment AB is parallel with segment CD [Given by definition]

*Segment AC is parallel with segment BD [Given by definition]

*We will have that the segment AD will be congruent with segment BC. [Opposite sides of a parallelogram are congruent]

*We then will have that segments AB will be congruent with AB, and segment DC will be congruent with DC. [Reflexive property.

*Then we will have:

`\begin{cases}\Delta DAB\cong\Delta CBA \\ \\ \Delta ADC\cong BCD\end{cases}`

[We will have this by SSS congruence]

*Then:

[Addition]

*Then we determine the value of each angle:

`\begin{cases}<\text{DAB}=90 \\ \\ <\text{ADC}=90\end{cases}`

[Division]

*Then we will have that:

*Then we have proved that a parallelogram with congruent diagonals is a rectangle.