In the previous post, we have proved the converse of the Pythagorean Theorem. In this post, we will prove that the diagonals of a rhombus are perpendicular to each other. That is, if we have parallelogram ABCD with diagonal and , then is perpendicular to .
What We Know
A rhombus is a parallelogram with four congruent sides. So, all sides of rhombus ABCD are congruent. That is
We also know that the diagonals of a parallelogram bisect each other. Since a rhombus is a parallelogram, it has also this property. Therefore, if point is the intersection of the diagonals as shown in the figure
What We Want to Show
Again, we want to show that is perpendicular to . Now, if we can show that , then, we will have proven the statement above.
From the given, we can see that
by reflexive property of congruence. A segment is congruent to itself.
So, by SSS congruence,
Now, we know that corresponding angles of congruent triangles are congruent. Since
and are corresponding angles
But since they are supplementary angles (Can you see why?)
Therefore, the diagonals of ABCD are perpendicular to each other, which is what we want to prove.