## Proof That The Opposite Sides of a Parallelogram Are Congruent

A parallelogram is a quadrilateral whose opposite sides are parallel. In the figure below, $PQRS$ is a parallelogram. $PQ$ is parallel to $RS$ and $PS$ is parallel to $QR$.

In this post, aside from being parallel, we will also prove that the opposite sides of a parallelogram are congruent.

## The Angle Secant Theorem

A secant is a line that intersects a circle at two points. In the figure below, $\angle E$ is formed by two secants. The angle intercepts two arcs $\overline{AB}$ and $\overline{CD}$. In this post, we will prove that the measure of the angle formed by two secants intersecting outside a circle is half the difference of the arcs intercepted by it.

To prove this theorem we will connect $\overline{BC}$ and use the Inscribed Angle Theorem and Exterior Angle Theorem.

Angle Secant Theorem

The angle measure formed by two secants intersecting outside a circle is half the difference of the arcs intercepted by it Continue reading

If $a > b > 0$, and $a^2 + b^2 = 6ab$, prove that
$\displaystyle\frac{a + b}{a - b} = \sqrt{2}$.