Lesson 5 ~25 min Unit 3 · Trigonometry +85 XP

Sides of a Right-Angled Triangle

Identify hypotenuse, opposite, and adjacent — the three sides as named with respect to a reference angle $\theta$. The opposite and adjacent SWAP when you change reference angle.

Today's hook: In the SAME triangle, moving your reference angle from one corner to the other swaps the opposite and adjacent. Why? Because ‘opposite’ and ‘adjacent’ depend on which angle you're looking at — only the hypotenuse stays fixed.

0/5QUESTS

Printable Worksheets

Print or save as PDF — or build a custom worksheet from any module's questions.

Build Apply Master Build custom

Think First

warm-up

In a right triangle, the side OPPOSITE an angle θ can never be the hypotenuse. Why? (Hint: the hypotenuse is opposite which angle?)

Record your answer in your workbook.

The Big Idea

+5 XP

Trigonometry needs new names for the sides of a right triangle. We label sides relative to a chosen angle $\theta$ (theta) — not relative to the right angle.

The three named sides are: Hypotenuse (still opposite the 90°, always longest), Opposite (across from $\theta$), and Adjacent (next to $\theta$, but NOT the hypotenuse). When the reference angle changes, opposite and adjacent swap. The hypotenuse never moves.

hyp = opposite 90° ; opp = opposite $\theta$ ; adj = next to $\theta$ (not hyp)

Hyp first

Always identify the hypotenuse first — it's opposite the right angle and the longest.

Opp = across

Opposite means “directly across from” the angle $\theta$.

Adj = the other one

Adjacent is the remaining leg — next to $\theta$ but not the hypotenuse.

What You'll Master

objectives

Know

The names hypotenuse, opposite and adjacent depend on $\theta$
Only the hypotenuse stays fixed
Opposite and adjacent swap when the reference angle changes

Understand

Why we need new names beyond 'leg' and 'hypotenuse'
How the same side can be 'opposite' for one angle and 'adjacent' for another
That the right angle is never the reference angle $\theta$

Can Do

Label hyp, opp, adj on a right triangle given a marked angle
Re-label sides if $\theta$ moves to a different vertex
Spot which side is which without confusion

Words You Need

vocabulary

$\theta$ (theta)The Greek letter used to label the reference angle in trigonometry. Always an angle other than the 90°.

HypotenuseThe side opposite the right angle. Always the longest side. Stays fixed when $\theta$ moves.

Opposite (opp)The side directly across from $\theta$ — the side $\theta$ does NOT touch.

Adjacent (adj)The side next to $\theta$ but not the hypotenuse. The leg $\theta$ touches.

Reference angleThe angle (not the right angle) being used for trig labelling. Symbol: $\theta$.

SwapWhen you change the reference angle to a different vertex, opposite and adjacent exchange roles.

Spot the Trap

heads-up

Wrong: “The opposite is always the vertical side.” No — it depends on where $\theta$ is.

Right: Opposite means across from $\theta$, no matter how the triangle is drawn.

Wrong: “The hypotenuse can be the adjacent.” Never — hyp is its own category.

Right: Hyp is fixed. Opp and adj are the two legs — whichever is across from $\theta$ is opposite.

Worked Labelling

+5 XP

Practise the three-step labelling: find the right angle → find $\theta$ → label hyp, opp, adj.

Step 1: Spot the small square — the side OPPOSITE it is the hypotenuse. Step 2: Find the reference angle marked $\theta$. Step 3: The side opposite $\theta$ is the opposite; the remaining leg is the adjacent.

Same triangle, two angles — opp and adj swap

Hyp first, always

Lock down the hypotenuse before anything else.

Stand at $\theta$

Imagine standing at $\theta$ and looking across — you see the opposite.

The leftover

Adj is the leg that's left after labelling hyp and opp.

Why the Swap?

+5 XP

In any right triangle, the two non-right angles are at different vertices. Moving $\theta$ from one to the other physically swaps which leg is across from $\theta$.

$\theta$ at vertex...	Opposite	Adjacent	Hypotenuse
$A$	$BC$	$AB$	$AC$
$C$	$AB$	$BC$	$AC$

(Right angle at $B$; $AC$ is the hypotenuse.) Notice $AB$ and $BC$ swap roles.

Hyp is fixed; opp and adj swap with $\theta$

Hyp is permanent

It always sits across from the 90°.

Legs swap

The two non-hyp sides exchange names when $\theta$ moves.

Two non-90° angles

In any right triangle there are exactly two angles you could call $\theta$.

Watch Me Solve It · 3 examples

Watch Me Solve It · Label sides with $\theta$ at one vertex

+15 XP per step

PROBLEM

In right triangle $ABC$ with the right angle at $B$: $AB = 3$, $BC = 4$, $AC = 5$. Mark $\theta$ at vertex $A$. Identify hyp, opp, adj.

1

Find hypotenuse

$AC = 5$ (longest, opposite the 90° at $B$).

The hypotenuse is always opposite the right angle.
2

Find opposite

Side across from $\theta$ (at $A$) is $BC = 4$.

$BC$ does not touch $A$.
3

Find adjacent

The remaining leg is $AB = 3$.

$AB$ touches both $A$ and $B$ — the leg next to $\theta$ that isn't hyp.

Answerhyp = $AC = 5$, opp = $BC = 4$, adj = $AB = 3$

Watch Me Solve It · Move $\theta$ to other vertex

+15 XP per step

PROBLEM

Same triangle ($AB = 3$, $BC = 4$, $AC = 5$, right angle at $B$). Now $\theta$ sits at vertex $C$. Re-identify hyp, opp, adj.

1

Hyp unchanged

$AC = 5$ still opposite the right angle.
2

New opposite

Across from $\theta$ (at $C$) is $AB = 3$.

$AB$ does not touch $C$.
3

New adjacent

Remaining leg = $BC = 4$.

Opp and adj swapped compared to the previous example!

Answerhyp = $AC = 5$, opp = $AB = 3$, adj = $BC = 4$

Watch Me Solve It · Labelling without numbers

+15 XP per step

PROBLEM

In right triangle $PQR$ with the right angle at $Q$, mark $\theta$ at $P$. Identify which side is the opposite.

1

Hyp

$PR$ (across from right angle at $Q$).
2

Opp

Across from $\theta$ (at $P$): $QR$.
3

Adj

Remaining: $PQ$.

Answeropp = $QR$

Common Pitfalls

heads-up

Treating opp/adj as fixed

Always calling the vertical side ‘opposite’ regardless of $\theta$.

Fix: Opp/adj move with $\theta$. Re-label every time the reference angle changes.

Confusing 90° with $\theta$

Marking the right angle as $\theta$.

Fix: $\theta$ is always one of the OTHER two angles, never the 90°.

Forgetting the hypotenuse rule

Calling a leg the hypotenuse just because it's the longest in the diagram.

Fix: Hyp is the side OPPOSITE THE RIGHT ANGLE — check the right-angle marker.

Copy Into Your Books

Three sides

Hyp = opposite 90°
Opp = across from $\theta$
Adj = next to $\theta$ (not hyp)

Reference angle

Symbol $\theta$
NOT the right angle
One of the two acute angles

Hyp is fixed

Doesn't move when $\theta$ moves
Always longest
Always opposite 90°

Opp/Adj swap

If $\theta$ changes vertex
Opp and adj exchange
Hyp stays put

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Side Labelling

4 problems

Four quick drills to lock in today's skill. Try each, then reveal the answer.

1 Which side is the hypotenuse?

The side opposite the right angle.Opposite the 90° (longest)
2 $\theta$ is at vertex $A$. Which side is ‘opposite’?

Side across from $\theta$ — the one that does not touch $A$.The side not touching $A$
3 In a 3-4-5 triangle with $\theta$ next to the side of length 4, which is adj?

Adj = leg next to $\theta$ but not hyp.The side of length 4
4 When $\theta$ moves to the other acute angle, what swaps?

Opposite and adjacent exchange names.Opp and adj

Complete in your workbook.

Quick Check · 5 questions

Which side is the hypotenuse?

+10 XP

In right $\triangle PQR$ with $\angle Q = 90°$ and $\theta$ at $P$, which is the opposite?

+10 XP

If $\theta$ moves to a different vertex, which side stays in the same name?

+10 XP

In a right triangle with legs 3 and 4 and hyp 5, if $\theta$ is at the angle facing the side of length 3, which is opp?

+10 XP

In a right triangle, $\theta$ is at one of the acute angles. Which of the following is NEVER possible?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyEasy2 MARKS

Q6. In right triangle $\triangle XYZ$ with the right angle at $Y$, $XY = 8$, $YZ = 15$, $XZ = 17$. If $\theta$ is at vertex $X$, identify the hyp, opp and adj.

Answer in your workbook.

UnderstandMedium3 MARKS

Q7. Using the same triangle from Q6, now $\theta$ moves to vertex $Z$. List the new hyp, opp and adj. State which sides swapped and which did not.

Answer in your workbook.

ReasonHard4 MARKS

Q8. In any right triangle, explain why the hypotenuse is the same side regardless of which acute angle is chosen as $\theta$, but the opposite and adjacent always swap when $\theta$ moves between the two acute angles.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. A — Opposite the 90°.

2. C — $QR$ is across from $P$.

3. A — Hyp is fixed.

4. B — By setup, the opposite is the side of length 3.

5. D — Hyp is never the adj.

Show Your Working Model Answers

Q6 (2 marks): Hyp = $XZ = 17$ (opposite right angle at $Y$) [1]. Opp $= YZ = 15$, adj $= XY = 8$ (opp is across from $X$, adj is the other leg) [1].

Q7 (3 marks): Hyp = $XZ = 17$ (unchanged) [1]. Opp $= XY = 8$, adj $= YZ = 15$ [1]. Opp and adj swapped; hyp stayed the same [1].

Q8 (4 marks): The hypotenuse is defined by its position relative to the RIGHT angle: opposite the 90° [1]. Since the right angle does not move, the hyp does not move [1]. Opposite is the side ACROSS from $\theta$, and adjacent is the OTHER leg [1]. Moving $\theta$ to the other acute angle physically changes which side is across from it, so opp and adj swap [1].

Stretch Challenge · +25 XP, +10 coins

Square the right way

In an isosceles right triangle (two equal legs $L$), find the ratio of the opposite side to the adjacent when $\theta$ is one of the acute angles. Then explain why this ratio is 1, no matter which acute angle you pick.

Reveal solution

Both legs equal $L$, so opp/adj $= L/L = 1$. Both acute angles are 45°, so the triangle is symmetric — opp and adj are the same length whichever acute angle is $\theta$.

Quick Review

Hypotenuse

Opposite the 90° — fixed

Opposite

Across from $\theta$

Adjacent

Next to $\theta$, not hyp

Three labels

hyp, opp, adj

$\theta$ is acute

Never the right angle

Swap rule

Opp & adj swap when $\theta$ moves

Your Badges

0 of 6

First Steps

3-Day Streak

3 in a Row

Lesson Ace

Stretch Seeker

Daily Warrior

Mark lesson as complete

Tick when you've finished Learn, Practice and the Stretch. Earns +85 XP and +25 coins.

Unit overview · Maths subject page