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 $\angle A \cong \angle C$ and $\angle B \cong \angle D$. To show that it is a parallelogram, we have to show that $AB \parallel CD$ and $AD \parallel BC$.

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

Proof

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

Congruent Opposite Sides of Quadrilaterals are Parallel

In this post, we are going to discuss one basic concepts of quadrilaterals. That is, if we have a quadrilateral, and the opposite sides are congruent, then these opposite sides are parallel.  This is the same if the two opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Consider the quadrilateral below. The lengths of AB and CD are equal. Also, The lengths of AD and BC are also equal. So, we are going to show that these pairs of sides are parallel.

Theorem: If the two pairs of the opposite sides of a quadrilateral are congruent, then they are parallel.

Given: $AB = CD$  and $AD = BC$Continue reading…

Proof by Contradiction: Odd and Its Square

We have had several examples on proof by contradiction in this blog like irrationality of square root of 3 and the sum of root of 2 and root of 3. In this post, we will have another detailed example.

In proof by contradiction, we want to change the statement “if P then Q” to “if NOT Q then NOT P.” These two statements are equivalent, so if we can prove the latter, then we have proved the former. Consider the following theorem.

Theorem: For any integer $x$, if $x^2$ is odd, then $x$ is odd.

We can assign the statements above to P (the hypothesis) and Q (the conclusion) as follows.

P: $x^2$ is odd

Q: $x$ is odd.

If we are going to change this statement to “if NOT Q then NOT P,” then, we have to find the opposite of P and opposite of Q.  Continue reading…