Labeling Sides Correctly: Opposite, Adjacent, Hypotenuse with Confidence

A reliable 5-step routine for labeling sides

Correctly labeling hypotenuse, opposite, and adjacent is not about where a side sits on the page (top/bottom/left/right). It depends only on two things: the right angle and your chosen reference angle (often written as θ).

Use this routine every time, in this exact order:

1. Locate the right angle (the 90° corner).
2. Mark the hypotenuse: the side across from the right angle (the longest side in a right triangle).
3. Circle the reference angle θ (one of the two non-right angles).
4. Label opposite: the side across from θ.
5. Label adjacent: the side that touches θ but is not the hypotenuse.

One quick self-check: at angle θ, exactly two sides touch it. One is the hypotenuse; the other is the adjacent.

Mini-glossary (angle-based, not position-based)

• Hypotenuse: opposite the right angle.
• Opposite (to θ): across from θ.
• Adjacent (to θ): touches θ and is not the hypotenuse.

Orientation practice: rotated and mirrored triangles

The goal is to label correctly even when the triangle is rotated, tilted, or mirrored. In each example, follow the 5 steps.

Example A: a “standard-looking” triangle (but still use the routine)

      C (right angle) ⟂
      |\
      | \
      |  \
      |   \
      A----B
        θ at A

• Step 1: Right angle is at C.
• Step 2: Hypotenuse is side AB (across from C).
• Step 3: Reference angle θ is at A.
• Step 4: Opposite to θ is side BC (across from A).
• Step 5: Adjacent to θ is side AC (touches A, not the hypotenuse).

Example B: rotated triangle (hypotenuse is not “on the bottom”)

        A
       /|
      / | 
     /  |  
    /   |   
   B----C
   right angle at B, θ at C

• Step 1: Right angle is at B.
• Step 2: Hypotenuse is side AC (across from B).
• Step 3: Reference angle θ is at C.
• Step 4: Opposite to θ is side AB (across from C).
• Step 5: Adjacent to θ is side BC (touches C, not the hypotenuse).

Notice: the “bottom” side here is BC, which happens to be adjacent to θ, but that is not a rule; it is just this drawing.

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Example C: mirrored triangle (adjacent is on the “other side”)

   A----B
    \   |
     \  |
      \ |
       \|
        C
 right angle at B, θ at A

• Step 1: Right angle is at B.
• Step 2: Hypotenuse is side AC (across from B).
• Step 3: Reference angle θ is at A.
• Step 4: Opposite to θ is side BC (across from A).
• Step 5: Adjacent to θ is side AB (touches A, not the hypotenuse).

Common pitfalls (and how to fix them fast)

Pitfall 1: Calling the side next to θ “adjacent” when it’s actually the hypotenuse

What goes wrong: You see a side touching θ and label it adjacent without first identifying the hypotenuse.

Fix: Always do Step 2 before Step 5. At θ, two sides touch. The one opposite the right angle is the hypotenuse; the other touching side is adjacent.

Quick check question: “Is this side across from the right angle?” If yes, it cannot be adjacent.

Pitfall 2: Mixing up opposite/adjacent when θ changes

What goes wrong: You label opposite/adjacent once, then keep those labels even after the reference angle moves to a different corner.

Fix: Hypotenuse stays the same (it depends on the right angle). Opposite and adjacent can swap when θ changes.

Pitfall 3: Assuming the bottom side is always adjacent

What goes wrong: You treat “adjacent” as “the base” because many textbook diagrams draw the triangle that way.

Fix: Adjacent is defined by touching θ and not being the hypotenuse. The adjacent side might be drawn on the left, right, top, or bottom depending on rotation.

One-sentence labeling drills (mixed orientations)

Directions: For each triangle, label H (hypotenuse), O (opposite to θ), and A (adjacent to θ). Then write one sentence explaining your choice (example sentence starter: “I know ___ is the hypotenuse because it is opposite the right angle at ___.”).

Drill 1

      P
     /|
    / | 
   /  |  
  Q---R
 right angle at Q, θ at R

• Your labels: H = ___, O = ___, A = ___
• One-sentence explanation: ______________________________

Drill 2 (mirrored)

  M---N
  |  /
  | /
  |/
  L
 right angle at M, θ at N

• Your labels: H = ___, O = ___, A = ___
• One-sentence explanation: ______________________________

Drill 3 (tilted)

      T
     / \
    /   \
   S_____U
 right angle at T, θ at S

• Your labels: H = ___, O = ___, A = ___
• One-sentence explanation: ______________________________

Drill 4 (θ changes; same triangle, new reference angle)

      A
     /|
    / | 
   /  |  
  B---C
 right angle at B

• Part (a): θ at A. Labels: H = ___, O = ___, A = ___. One sentence: ______________________________
• Part (b): θ at C. Labels: H = ___, O = ___, A = ___. One sentence: ______________________________

Drill 5 (spot the pitfall)

   D
   |\
   | \
   |  \
   E---F
 right angle at E, θ at F

A student says: “Side DF is adjacent because it touches θ at F.”

• Is the student correct? Yes / No
• Correct labels: H = ___, O = ___, A = ___
• One-sentence correction: ______________________________