Proof that The Diagonals of a Rhombus are Perpendicular

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 $latex \overline{AC}$ and $latex \overline{BD}$, then $latex \overline{AC}$ is perpendicular to $latex \overline{BD}$.

parallelogram 2

What We Know

A rhombus is a parallelogram with four congruent sides. So, all sides of rhombus ABCD are congruent. That is

$latex \overline{AB} \cong \overline{BC} \cong \overline{CD} \cong \overline{DA}$.  Continue reading

Quadrilaterals With Congruent Opposite Angles Are Parallelogram

In the previous post, we have proved that if the opposite sides of a quadrilateral are congruent, then they are parallel. In this post, we are going to show that if the opposite angles of a quadrilateral are congruent, then it is a parallelogram.

In the figure below, we have quadrilateral ABCD with $latex \angle A \cong \angle C$ and $latex \angle B \cong \angle D$. To show that it is a parallelogram, we have to show that $latex AB \parallel CD$ and $latex AD \parallel BC$.

Theorem: If two pairs of opposite angles of a quadrilateral are congruent, then it is a parallelogram.



It is given that $latex \angle A \cong \angle C$ and $latex \angle B \cong \angle D$.  Continue reading

Proof that the Product of Two Rational Numbers is Rational

The purpose of this blog is to introduce mathematical proofs as early as possible. When I say mathematical proof, it does not have to be rigorous; sometimes simple reasoning by words alone would suffice especially in the early grades.

In this post, I am going to show you a simple proof that can be probably given as a problem for middle school students. That is, when the students already understand the definition (and meaning) of rational numbers.

A rational number is a number which can be expressed as a fraction whose numerator and denominator are both integers and the denominator is not equal to 0. A student will probably argue that the fractions of integral numerators and denominators, but it is not the case. The fraction $latex \frac{\pi}{2}$ (not the image above) for instance is a fraction, but not a rational number. From that definition, we can show that the product of two rational numbers is rational. I have written the proof as layman as possible. Continue reading