The angle adjacent to the run and opposite to the rise is called the slope angle.

We denote an angle of interest with a symbol. The Greek letter \(\theta\) (pronounced theta) is frequently used.

We can abbreviate hypotenuse, opposite, and adjacent to hyp, opp, and adj, respectively.

Common Practice

• In equations, write “tan” for tangent.
• Say “tan” when referring to the tangent ratio.
• Write brackets around the angle.

​​​​​Note that \(\tan (\theta )= \dfrac{\text{opp}}{\text{adj}} = \dfrac{1}{2}\) for each triangle.

A tangent ratio of \(\dfrac{1}{2}\) can be achieved in multiple ways.

Why is this?

As long as rise and run are in a \(1\) to \(2\) ratio, the tangent ratio simplifies to \(\dfrac{1}{2}\) regardless of the size of the triangle.

All of the triangles with this tangent ratio were similar to one another (i.e., the triangles will have the same shape and the corresponding side lengths will be in the same proportions).

Solution

Given \(\triangle ABC \sim \triangle EDF\), we know that \(\angle B\) and \(\angle D\) are corresponding, and therefore equal.

Therefore, \(\tan(D) = \dfrac{4}{3}\).