Angle measurement — protractor use, angle relationships (complementary, supplementary): Worked examples

This page is the practice ground for Angle Measurement. Before we start, one promise: every symbol you see here is explained the moment it appears. If a word like "vertex" or "complement" shows up, it was already built on the parent note — but we will re-anchor it to a picture so you never have to flip back.

We use only two ideas, and both are just sentences turned into equations:

The scenario matrix

Every problem this topic can throw at you falls into one of the cells below. The worked examples that follow are each tagged with the cell they cover, so by the end no cell is left unshown.

#	Case class	What makes it tricky	Example
C1	Protractor read — baseline on the left	must pick the correct 0° scale	Ex 1
C2	Protractor read — baseline on the right	the other scale is the correct one	Ex 2
C3	Plain complement / supplement	which total, 90 or 180?	Ex 3
C4	Degenerate / boundary value (0°, 90°, 180°)	does the angle even have a partner?	Ex 4
C5	Ratio split of a total	translate "a:b" into parts	Ex 5
C6	Word equation ("more than / twice")	turn English into an equation	Ex 6
C7	Real-world word problem	strip the story down to angles	Ex 7
C8	Exam twist — angles built from an unknown $x$	solve, then reject impossible answers	Ex 8
C9	Reflex angle (> 180°) on a 180° protractor	the tool can't read it directly	Ex 9

The worked examples

[!example] Example 1 — C1: reading a protractor with the baseline on the LEFT

Statement: A protractor is laid so one ray points left along its baseline. The second ray crosses the curved edge. Two numbers sit near the crossing: the outer scale says 150°, the inner scale says 30°. What is the true angle?

Forecast: Guess now — is the angle acute (small, pointy) or obtuse (wide, open)? Write down 30 or 150 before reading on.

Figure — Angle measurement — protractor use, angle relationships (complementary, supplementary)

Steps:

Find where our baseline ray meets 0°. Because the ray points left, its 0° mark is the outer scale's zero (recall the outer ring counts up left-to-right).
- Why this step? The rule from the parent note: "start from the 0° mark of your baseline ray and count up." You must first locate which scale reads zero on your ray.
Follow that same (outer) scale up to the second ray. Look at the red arc in the figure — it sweeps along the outer numbers and lands on 150°.
- Why this step? A scale is only consistent if you read the whole angle on the one that started at your zero. Jumping scales mid-way is Mistake 1.
The angle is $150°$ .

Verify: Is it obtuse? $150° > 90°$ ✓ and the drawn opening is clearly wide, matching an obtuse angle. The distractor 30° was the mirror scale (note $180° - 150° = 30°$ ).

[!example] Example 2 — C2: reading a protractor with the baseline on the RIGHT

Statement: Same protractor, but now the baseline ray points right. The crossing shows outer = 115°, inner = 65°. Find the true angle.

Forecast: Which scale wins this time — inner or outer?

Steps:

The right-pointing ray hits 0° on the inner scale (the inner ring counts up right-to-left, so its zero is on the right).
- Why this step? Same rule as Ex 1 — but the side your ray sits on flips which scale carries the zero.
Follow the inner scale up to the second ray: it reads 65° (green arc in the figure).
- Why this step? Consistency again: zero was inner, so the reading must be inner. Grabbing the outer 115° here would be Mistake 1 in action.
The angle is $65°$ .

Verify: $65° < 90°$ , so acute — the drawn opening is narrow ✓. The outer distractor is $180° - 65° = 115°$ , exactly the "wrong scale" trap.

[!example] Example 3 — C3: a plain complement and a plain supplement

Statement: Call our known angle $∠ P$ (the symbol $∠$ means "angle," and $P$ is just the name we give this particular one — pick any letter). We are told $∠ P = 42°$ . Find (a) its complement and (b) its supplement.

Forecast: Two answers are coming. Which one is the bigger number — the complement or the supplement?

Steps:

Complement means "adds to 90°," so complement $= 90° - 42°$ .
- Why this step? Complement fills a corner (right angle). Subtracting the known angle from the total leaves the partner.
$90° - 42° = 48°$ .
Supplement means "adds to 180°," so supplement $= 180° - 42°$ .
- Why this step? Supplement fills a straight line. Different total, so use 180°. Mixing these two totals up is Mistake 2.
$180° - 42° = 138°$ .

Verify: $42° + 48° = 90°$ ✓ and $42° + 138° = 180°$ ✓. The supplement (138°) is larger, as expected — 180 is a bigger total than 90.

[!example] Example 4 — C4: the degenerate / boundary cases

Statement: Decide, for each, whether the partner is a proper angle (one you can actually draw as an opening) or a degenerate $0°$ "angle" (two rays lying exactly on top of each other): (a) complement of $90°$ , (b) complement of $0°$ , (c) supplement of $180°$ .

Forecast: At least one of these has NO proper partner — its partner collapses to a zero-width $0°$ . Which do you think collapses?

Steps:

Complement of $90°$ : $90° - 90° = 0°$ .
- Why this step? The formula still runs, but a $0°$ result is two rays lying on top of each other — a degenerate angle, not a real opening. So $90°$ has no proper complement.
Complement of $0°$ : $90° - 0° = 90°$ .
- Why this step? Here the partner is a full, proper $90°$ angle — the largest a complement can ever be. This one is fine.
Supplement of $180°$ : $180° - 180° = 0°$ .
- Why this step? A straight angle already fills the whole line, so its supplement collapses to the degenerate $0°$ — again no proper partner.

Verify: $90 + 0 = 90$ ✓, $0 + 90 = 90$ ✓, $180 + 0 = 180$ ✓. So cases (a) and (c) give the degenerate $0°$ (no proper partner), while (b) gives a genuine $90°$ — matching the forecast that some cases collapse. Boundary rule that emerges: complements live in $[0°, 90°]$ , supplements in $[0°, 180°]$ ; at the very ends the partner is squeezed to $0°$ .

[!example] Example 5 — C5: splitting a total by a ratio

Statement: Two supplementary angles are in the ratio $4 : 5$ . Find both.

Forecast: The parts add to how many "chunks" — 9 or 20?

Steps:

Call the angles $4 x$ and $5 x$ , where $x$ is one "chunk" in degrees.
- Why this step? A ratio $4 : 5$ means "4 chunks to 5 chunks." Letting $x$ be one chunk turns the story into algebra. See Angle Addition Postulate for why the two pieces legitimately add.
Supplementary ⇒ they sum to 180°: $4 x + 5 x = 180$ .
- Why this step? Applying the definition; the total is the equation.
Combine: $9 x = 180$ , so $x = 20$ .
- Why this step? $4 + 5 = 9$ chunks share the 180°; dividing gives one chunk.
First angle $= 4 x = 80°$ ; second $= 5 x = 100°$ .

Verify: Ratio $80 : 100 = 4 : 5$ ✓; sum $80 + 100 = 180°$ ✓.

[!example] Example 6 — C6: an English word-equation

Statement: An angle is twice its complement. Find the angle.

Forecast: Bigger or smaller than 45°? (An angle equal to its complement is exactly 45°.)

Steps:

Let the angle be $x$ ; its complement is $(90 - x)$ .
- Why this step? Complement $= 90° -$ angle, always.
Translate "the angle is twice its complement": $x = 2 (90 - x)$ .
- Why this step? "Twice" means multiply the complement by 2; "is" becomes the equals sign.
Expand: $x = 180 - 2 x$ .
- Why this step? Distributing the 2 clears the bracket so we can gather $x$ terms.
Add $2 x$ to both sides: $3 x = 180$ , so $x = 60°$ .
- Why this step? Collecting all $x$ on one side isolates the unknown.

Verify: Complement $= 90 - 60 = 30°$ ; is $60 = 2 \times 30$ ? Yes ✓. And $60° > 45°$ matches the forecast (it's the bigger of a pair).

[!example] Example 7 — C7: a real-world problem

Statement: A folding laptop screen opens from the keyboard. The keyboard lies flat (a straight surface). The screen makes a $115°$ angle with the front half of the keyboard. What angle does the screen make with the back edge of the keyboard behind the hinge?

Forecast: The two angles share the flat surface — do they add to 90° or 180°?

Steps:

The keyboard surface is one straight line through the hinge (the vertex). The screen is a ray from that hinge.
- Why this step? A straight surface = a straight angle = 180°. Naming the vertex tells us the two angles share it. This is a linear pair — two angles on a straight line.
The front angle ( $115°$ ) and the back angle sit on that straight line, so they are supplementary: back $= 180° - 115°$ .
- Why this step? Supplement fills the straight line; subtract the known part.
Back angle $= 65°$ .

Verify: $115° + 65° = 180°$ ✓ — together they lie flat along the keyboard, exactly a straight line. Units are degrees throughout ✓.

[!example] Example 8 — C8: the exam twist (reject the impossible)

Statement: Two angles are complementary. One is $(2 x + 10) °$ and the other is $(3 x - 20) °$ . Find $x$ and both angles — and check that both are valid angles.

Forecast: Could a "complementary" angle secretly come out negative? Watch for it.

Steps:

Complementary ⇒ sum is 90°: $(2 x + 10) + (3 x - 20) = 90$ .
- Why this step? Definition first; the total becomes the equation.
Combine like terms: $5 x - 10 = 90$ .
- Why this step? $2 x + 3 x = 5 x$ and $10 - 20 = - 10$ ; tidying prepares to solve.
Add 10, then divide by 5: $5 x = 100 \Rightarrow x = 20$ .
- Why this step? Standard isolation of the unknown.
Substitute back: first $= 2 (20) + 10 = 50°$ ; second $= 3 (20) - 20 = 40°$ .
- Why this step? An answer for $x$ is only useful once turned back into the actual angles.
Validity check: both $50°$ and $40°$ are positive and each $< 90°$ — genuine acute angles, so nothing is rejected.
- Why this step? The exam twist is that a negative or over-90° result would be geometrically impossible; you must confirm it survives.

Verify: $50° + 40° = 90°$ ✓, both positive ✓, both acute ✓.

[!example] Example 9 — C9: measuring a REFLEX angle (> 180°) with a 180° protractor

Statement: A reflex angle is one bigger than 180° but less than 360° — the "big way round" between two rays. Your protractor only reads up to $180°$ , so it cannot measure a reflex angle directly. The two rays enclose a small opening you can measure: it reads $130°$ . What is the reflex angle on the other side?

Forecast: A full turn is 360°. If the small opening is 130°, is the reflex side more or less than 230°?

Steps:

Measure the ordinary (non-reflex) angle with the protractor the usual way: it is $130°$ .
- Why this step? The protractor tops out at 180°, so we measure the part it can reach — the smaller opening between the rays.
The small opening and the reflex opening together make one full turn = $360°$ .
- Why this step? Sweeping all the way around from one ray back to itself is a complete rotation. The two openings are the two "ways round," so they add to 360°.
Reflex angle $= 360° - 130° = 230°$ .
- Why this step? Subtract the measured part from the full turn to get the leftover big part.

Verify: $130° + 230° = 360°$ ✓ (one full rotation), and $230° > 180°$ ✓ so it genuinely is reflex. Sanity check on the forecast: yes, more than the small side.

Recall Quick self-test (reveal after answering)

Complement of an angle uses which total? ::: $90°$ (fills a right-angle corner). Supplement of an angle uses which total? ::: $180°$ (fills a straight line). If the baseline ray reads 0° on the outer scale, which scale do you read to the end? ::: The outer scale — follow your zero's flow. Two supplementary angles in ratio $4 : 5$ are? ::: $80°$ and $100°$ . An angle twice its complement is? ::: $60°$ . A reflex angle whose small side is $130°$ measures? ::: $360° - 130° = 230°$ .

Related builders: Vertical Angles, Parallel Lines and Transversals, Triangle Angle Sum, and later Trigonometry Basics, Coordinate Geometry, Circle Theorems.