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Right-Triangle Trigonometry Essentials: What Makes a Triangle “Right” and Why Ratios Matter

Capítulo 1

Estimated reading time: 4 minutes

What Is a Right Triangle?

A right triangle is a triangle with one angle measuring exactly 90°. That 90° angle is called the right angle. The two sides that form the right angle meet like the corner of a square.

How to identify the right angle

Look for a small square marking in the corner (common in diagrams).
If angles are labeled, find the one that equals 90°.
The right angle is always formed by the two shorter sides (not the longest side).

Why Ratios Matter (What Trigonometry Is Doing Here)

In this course, right-triangle trigonometry is built on one simple idea: compare side lengths using ratios. A ratio is just a fraction made from two lengths, like 3/5 or 7/10.

Why ratios? Because if two right triangles have the same acute angle, their shapes match (they are similar), and the side-length ratios stay the same even if the triangles are different sizes. That means trigonometry is not about memorizing random rules—it’s about using consistent comparisons between sides.

A quick ratio reminder

If a triangle is scaled up (every side multiplied by the same number), then ratios don’t change:

Small triangle sides: 3 and 4  → ratio = 3/4 = 0.75  Large triangle sides: 6 and 8  → ratio = 6/8 = 0.75

This “ratio stays the same” idea is the backbone of sine, cosine, and tangent later.

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The Hypotenuse: Your First Quick Check

The hypotenuse is the side with a special job in every right triangle.

Quick checks for the hypotenuse

Always across from the right angle. If you find the 90° angle, the side directly opposite it is the hypotenuse.
Always the longest side. No other side can be longer than the hypotenuse in a right triangle.

These checks work no matter how the triangle is rotated or drawn.

Reference Angle (θ): How “Opposite” and “Adjacent” Are Chosen

Besides the right angle, a right triangle has two acute angles (each less than 90°). In trigonometry, we pick one of those acute angles to focus on. That chosen angle is the reference angle, often labeled θ (theta).

Once θ is chosen, the names of the other sides are determined relative to θ:

Opposite: the side directly across from θ.
Adjacent: the side next to θ that is not the hypotenuse.
Hypotenuse: still across from the right angle (this one never changes).

Step-by-step: Labeling sides from a chosen θ

Find the right angle (90°).
Label the side across from 90° as the hypotenuse. (Quick check: it should be the longest.)
Locate θ. (One of the acute angles.)
Opposite is across from θ.
Adjacent is the side touching θ that is not the hypotenuse.

Key idea: opposite/adjacent depend on θ

If you choose a different acute angle as θ, the opposite and adjacent sides swap roles, but the hypotenuse stays the same.

Side name	How it’s determined	Changes if θ changes?
Hypotenuse	Across from the 90° angle; longest side	No
Opposite	Across from θ	Yes
Adjacent	Next to θ but not the hypotenuse	Yes

Mini Practice: Label Hypotenuse / Opposite / Adjacent

For each exercise, do the step-by-step labeling process. Write H for hypotenuse, O for opposite (to θ), and A for adjacent (to θ).

Exercise 1 (θ at the bottom-left)

A right triangle has vertices A, B, C. Angle C is 90°. Angle A is labeled θ. Sides are AB, BC, AC.

Which side is the hypotenuse?
Which side is opposite θ?
Which side is adjacent to θ (not the hypotenuse)?

Exercise 2 (θ moved to the other acute angle)

Same triangle as Exercise 1: angle C is 90°. Now angle B is labeled θ.

Which side is the hypotenuse?
Which side is opposite θ?
Which side is adjacent to θ (not the hypotenuse)?

Exercise 3 (different labeling, rotated drawing)

A right triangle has vertices P, Q, R. Angle Q is 90°. Angle R is labeled θ. Sides are PQ, QR, PR. The triangle is drawn rotated so the hypotenuse is not horizontal.

Identify the hypotenuse using the “across from 90°” check.
Identify the opposite side to θ.
Identify the adjacent side to θ (not the hypotenuse).

Exercise 4 (quick-check focus)

A right triangle has a clearly marked right angle at vertex M. The sides are MN, MP, and NP. Angle N is labeled θ.

First, label the hypotenuse (longest side, across from the right angle).
Then label opposite and adjacent relative to θ at N.

Now answer the exercise about the content:

In a right triangle, what stays the same and what can change when you choose a different acute angle as the reference angle θ?

You are right! Congratulations, now go to the next page

You missed! Try again.

The hypotenuse is always across from the 90° angle, so it never changes. The opposite and adjacent sides are defined relative to θ, so picking a different θ can swap which side is opposite and which is adjacent.