Quarry School

Right triangles and the names of their sides

Explain it like I am five

Lean a 17-foot ladder on a wall: bottom 8 feet from the wall, top 15 feet up. Floor and wall meet in a square 90° corner, so the three form a right triangle, a triangle with one 90° corner. The ladder faces that corner: it is the hypotenuse (hy-POT-uh-noos), the longest side. The floor and wall pieces are the legs.

Stand where the ladder meets the floor and call the angle there θ (theta, a Greek letter used as an angle's name). The floor leg, 8, touches your corner: it is adjacent, which means next to. The wall leg, 15, sits across from you: it is opposite. At the top corner the two names swap: 15 is adjacent, 8 is opposite. The ladder stays the hypotenuse.

Why: the Pythagorean theorem says leg2 + leg2 = hypotenuse2 (82 means 8 × 8): 64 + 225 = 289 = 172, so 17 beats each leg. The three angles add to 180° and the square corner takes 90°, so the two pointed corners share the other 90°: each is acute, smaller than 90°.

In plain words

Picture a ladder leaning against a wall. The floor and wall make a square corner, exactly 90°. A triangle containing that corner is a Right triangle. The ladder faces the square corner and is the Hypotenuse (hyp), the longest side. The other two sides are legs. Pick one pointed corner and call its angle θ, read theta. The leg facing your corner is the Opposite side (opp). The leg touching your corner is the Adjacent side (adj). Move to the other pointed corner and those two names trade places. The hypotenuse stays the same. Your chosen angle is an Acute angle, smaller than a square corner.

θadjopphyp
Standing at θ: the bottom leg touches you (adjacent), the upright leg faces you (opposite), and the slanted side faces the square corner (hypotenuse).
Reminder
  • Angle letters. θ means theta and names an angle. A and B can name angles too; the names do not tell you their sizes.
  • Squaring. 32 means 3 × 3 = 9. In a 3, 4, 5 right triangle, 32 + 42 = 52.
  • Acute angle. 0° < θ < 90° means θ lies strictly between 0° and 90°; 40° is acute and 90° is not.
Why it works. The three angles of every triangle add up to 180°. The square corner uses 90° of that, so the other two angles share the remaining 90°. Each one is therefore smaller than 90°, which is what acute means. That is why this first definition only covers acute angles: a right triangle has no room for a second angle of 90° or more. The hypotenuse is the longest side because the Pythagorean theorem gives c2 = a2 + b2, which is bigger than a2 alone and bigger than b2 alone.
RuleFor an acute angle θ of a right triangle: hyp faces the 90° corner; opp is the leg facing θ; adj is the leg touching θ. The hypotenuse is excluded when naming adj.
The two acute angles add to 90°.
The same idea, five ways
Say it

Hypotenuse faces the square corner. Opposite faces your chosen angle. Adjacent is the leg beside it.

Write it

Opposite and adjacent depend on the chosen angle; the hypotenuse does not.

In math
  • hyp: across from 90°
  • opp: across from θ
  • adj: touches θ and is not hyp
  • A + B = 90°
Like

Your left and right change when you turn around; the building beside you stays the same.

See it
θadjopphyp
Standing at θ: the bottom leg touches you (adjacent), the upright leg faces you (opposite), and the slanted side faces the square corner (hypotenuse).
The same idea, other ways
The ladder story

Stand at the ladder's foot. The ladder is hyp, the wall you face is opp, and the floor beside you is adj. Imagine standing at the top instead: the wall is now beside you and the floor faces you.

θfloorwallladder
At the floor angle, the wall faces you and the floor touches you.
Trace the corners

Find the square corner first and trace the side that never touches it: hyp. Put a finger on θ. Of the two remaining sides, one touches your finger, adj, and one does not, opp.

θadjopphyp
Standing at θ: the bottom leg touches you (adjacent), the upright leg faces you (opposite), and the slanted side faces the square corner (hypotenuse).
Same sides, different viewpoints

For a triangle with sides 3, 4 and 5, one pointed corner sees opp = 3 and adj = 4. The other sees opp = 4 and adj = 3. The side 5 faces the square corner in both views.

θ345
Redrawing the same triangle from the other acute angle swaps the two leg labels.
.1Hypotenuse (hyp)

The hypotenuse is the ladder in the wall and floor picture. It is the only side that does not touch the square corner. Its name stays fixed when you choose another acute angle.

  • hyp faces 90°.
  • hyp is the longest side.
θ12915
The side 15 faces the square corner.
The same idea, five ways
Say it

Hypotenuse is the side across from the right angle.

Write it

The hypotenuse is the side that faces the 90° corner.

In math
  • hyp faces 90°.
  • hyp is the longest side.
Like

The ladder stays the ladder whether you stand at its foot or top.

See it
θ12915
The side 15 faces the square corner.
Worked exampleFind the side whose name stays fixed

A right triangle has legs 9 and 12 and side 15 facing the square corner. Find hyp. In words, find the side opposite 90°.

θ12915
Name the sides from the marked angle before making a fraction.
  1. Find the marked 90° corner.The right angle determines the hypotenuse.
  2. The side facing it is 15, so hyp = 15.It is the only side that does not meet that corner.
Answer
hyp = 15.
Check 92 + 122 = 81 + 144 = 225 = 152, and 15 exceeds both legs.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Choose hyp by the direction it slants.
Rotating the page changes that direction without changing the triangle.
✓ Instead: Find the square corner and look across it.
Tips and tricks
  • The little square corner is your anchor.
.2Opposite side (opp)

Opposite means across from where you stand. Put your finger on the requested acute angle. The opposite leg is the one your finger cannot touch without crossing the triangle.

  • opp faces θ.
  • opp is a leg, not hyp.
θ12915
From the corner beside side 12, leg 9 is across the triangle.
The same idea, five ways
Say it

Opposite is the leg across from theta.

Write it

The opposite side is the leg that does not touch the chosen acute angle.

In math
  • opp faces θ.
  • opp is a leg, not hyp.
Like

It is like the wall across from you when you stand at the ladder's foot.

See it
θ12915
From the corner beside side 12, leg 9 is across the triangle.
Worked exampleFind the leg across from your corner

θ is between side 12 and hypotenuse 15. Find opp. In words, find the leg that does not meet θ.

θ12915
Name the sides from the marked angle before making a fraction.
  1. The sides meeting θ are 12 and 15.They form the chosen corner.
  2. The remaining side is 9, so opp = 9.It faces θ and does not touch it.
Answer
opp = 9.
Check At the other acute corner side 9 touches that angle and becomes adjacent, confirming that the name depends on the angle.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: The longest side is opposite θ.
The longest side faces 90°, not the acute θ.
✓ Instead: opp for an acute θ is a leg.
Tips and tricks
  • Point to θ before naming opp.
.3Adjacent side (adj)

Adjacent means next to where you stand. Two sides touch θ, but one already has the name hypotenuse. Adjacent is the other touching side, the leg beside θ.

  • adj touches θ.
  • adj is the touching leg, not hyp.
θ12915
Side 12 touches θ and is a leg.
The same idea, five ways
Say it

Adjacent is the leg next to theta.

Write it

The adjacent side is the leg touching the chosen angle, excluding the hypotenuse.

In math
  • adj touches θ.
  • adj is the touching leg, not hyp.
Like

It is like the floor beside you at the ladder's foot.

See it
θ12915
Side 12 touches θ and is a leg.
Worked exampleChoose between the two touching sides

θ is between side 12 and hypotenuse 15. Find adj. In words, choose the touching side that is a leg.

θ12915
Name the sides from the marked angle before making a fraction.
  1. Set side 15 aside as hyp.It faces the square corner and cannot be renamed adjacent.
  2. The other touching side is 12, so adj = 12.Adjacent is the touching leg.
Answer
adj = 12.
Check opp = 9 and adj = 12 are the two legs, while hyp = 15 is the third side; every side has one role.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: adj = 15 because it touches θ.
That side is the hypotenuse, which the adjacent-leg label excludes.
✓ Instead: adj = 12.
Tips and tricks
  • Adjacent must touch θ and be a leg.
Strategy: step by step
  1. Find the square corner. The side across from it is the hypotenuse; label it hyp.
  2. Put your finger on the angle the problem asks about.
  3. Of the two legs, the one touching that angle is adj.
  4. The leg that does not touch that angle is opp.
  5. If the problem switches to the other acute angle, swap opp and adj. The hyp stays the same.
Strategy
Name the sides from the chosen angle
1
Does the triangle have a 90° corner?
YesLabel the side facing it hyp.
NoDo not apply right-triangle side ratios.
↓
2
Is the touching side the hypotenuse?
YesRetain hyp; the other touching side is adj.
NoIf it is a leg, it is adj.
↓
3
Did the chosen acute angle change?
YesSwap opp and adj, retain hyp.
NoKeep the labels.
  1. Find the square corner. The side across from it is the hypotenuse; label it hyp.
  2. Put your finger on the angle the problem asks about.
  3. Of the two legs, the one touching that angle is adj.
  4. The leg that does not touch that angle is opp.
  5. If the problem switches to the other acute angle, swap opp and adj. The hyp stays the same.
Worked exampleName the sides from both acute angles

A right triangle has legs 3 and 4 and hypotenuse 5. Angle A is the corner where the leg 4 meets the hypotenuse. Angle B is the corner where the leg 3 meets the hypotenuse. Name opp, adj and hyp for A, then for B. In words, label the same sides from two different corners.

θ435
From A, leg 4 touches the chosen corner.
θ345
Redraw from B with the chosen angle at θ: the two leg names swap.
  1. The legs 3 and 4 meet at the square corner, and the side across from it is 5. So hyp = 5 for both angles.The hypotenuse faces the right angle, whichever acute angle you stand at.
  2. Stand at A. The leg touching A is 4, so adj = 4.Adjacent means next to: it is one of the two sides that form the corner at A, and it is not the hypotenuse.
  3. The leg across from A is 3, so opp = 3.It is the one side that does not touch A.
  4. Stand at B. Now the leg 3 touches you and the leg 4 is across: adj = 3, opp = 4, hyp = 5.Moving to the other acute angle swaps opposite and adjacent.
Answer
  • At A:
  • opp = 3
  • adj = 4
  • hyp = 5
  • At B:
  • opp = 4
  • adj = 3
  • hyp = 5
Check 32 + 42 = 9 + 16 = 25 = 52, so 5 is the hypotenuse, and it is the longest side, as it must be.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: The hypotenuse touches θ, so call it adj.
Adjacent means the touching leg. The hypotenuse has its own name because it faces 90°.
✓ Instead: Name hyp first, then choose adj from the two legs.
✗ Not this: The vertical leg is always opposite.
A side's direction on the page does not tell you which angle the problem chose.
✓ Instead: Point to θ. The leg that does not touch it is opp.
✗ Not this: Use a 110° corner as θ inside a right triangle.
The 90° corner leaves only 90° for the other two positive angles together.
✓ Instead: Use triangle ratios for an acute corner; later definitions handle larger angles.
Tips and tricks
  • Hypotenuse first, angle second, two legs last.
  • Say across for opposite and next to for adjacent while pointing.
  • Memory cue: H faces the right angle; O faces your angle; A is the remaining touching leg.
Trap. Calling the hypotenuse the adjacent side. The hypotenuse also touches θ, but it is never called adjacent. Name the hypotenuse first, set it aside, and only then decide which of the two legs touches θ.
Keep in mind
  • Name the hypotenuse first and set it aside: it faces the 90° corner, so in a triangle with sides 8, 15 and 17 it is 17 from either pointed corner.
  • Opposite and adjacent depend on your corner, not on the page: the 15-foot wall leg is opposite the angle at the ladder's bottom but adjacent to the angle at its top.
  • The hypotenuse touches your angle too, yet it is never called adjacent: adjacent always means the leg that touches your corner, as the 8-foot floor leg does for the angle at the ladder's bottom.
Memory hookAdjacent adjoins you, opposite is over there, and the hypotenuse is huge: the longest side, facing the square corner.
Flash cards: say the answer out loud, then flip
What is a right triangle?
A triangle with one 90° (square) corner.
What is the hypotenuse?
The side across from the 90° corner. It is always the longest side.
What does adjacent mean?
Next to: the leg that touches your angle, never the hypotenuse.
A right triangle has sides 9, 40 and 41. Which one is the hypotenuse?
41, because 92 + 402 = 81 + 1600 = 1681 = 412.
One acute angle of a right triangle is 28°. What is the other one?
62°, because the two acute angles add to 90°.
From angle A, opp = 8, adj = 15 and hyp = 17. From the other acute angle B, is opp still 8?
No. Moving to the other corner swaps the legs: from B, opp = 15 and adj = 8. The hypotenuse stays 17.