Identify opposite, adjacent, and hypotenuse sides

In a right triangle, the hypotenuse is the side opposite the right angle and is always the longest side; relative to a selected acute angle, the opposite side lies across from that angle, while the adjacent side is the non-hypotenuse side that forms the angle. These roles depend on the chosen angle and provide the geometric basis for sine, cosine, and tangent; the classification is limited to right triangles, not oblique triangles or generalized angle conventions.

Detailed Explanation: Identify opposite, adjacent, and hypotenuse sides

In a right triangle, identify the sides in this order:

Find the right angle.
The side directly across from the $90^\circ$ angle is the hypotenuse. It is always the longest side.
Choose the acute angle you are using.
The side directly across from that angle is the opposite side.
The remaining side that touches the chosen angle, but is not the hypotenuse, is the adjacent side.

Worked example

Consider right triangle $ABC$ , where $\angle C=90^\circ$ . Suppose we are identifying the sides relative to $\angle A$ .

The side across from $\angle C$ is $\overline{AB}$ , so $\overline{AB}$ is the hypotenuse.
The side across from $\angle A$ is $\overline{BC}$ , so $\overline{BC}$ is the opposite side.
The side that touches $\angle A$ and is not the hypotenuse is $\overline{AC}$ , so $\overline{AC}$ is the adjacent side.

Therefore, relative to $\angle A$ :

\text{hypotenuse}=\overline{AB},\qquad \text{opposite}=\overline{BC},\qquad \text{adjacent}=\overline{AC}.

Remember that opposite and adjacent depend on the angle selected. If you used $\angle B$ instead, $\overline{AC}$ would be opposite and $\overline{BC}$ would be adjacent. The hypotenuse would remain $\overline{AB}$ .

Learn by doing: Identify opposite, adjacent, and hypotenuse sides

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Trigonometry - Labeling of Sides, Reversed

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