1.2.3 · D5Basic Geometry
Question bank — Angle measurement — protractor use, angle relationships (complementary, supplementary)
Recall Angle-type reminder (used all over this page)
Zero angle ::: exactly — rays overlap, no turn (degenerate). Acute angle ::: strictly between and . Right angle ::: exactly — the "corner". Obtuse angle ::: strictly between and . Straight angle ::: exactly — rays form one line (degenerate). Reflex angle ::: strictly between and — the "big" turn on the outside. Full detail lives in ::: Types of Angles.
The chart above is the reference for every "acute / obtuse / reflex" verdict in this bank — glance back at the coloured wedges whenever a question names a type.
True or false — justify
The complement of an angle is always smaller than the angle itself.
False — it depends on the angle. If the complement is larger; only when is the complement smaller. At exactly both equal .
Every angle has a complement.
False — an angle needs measure for to be a non-negative angle. A angle has no complement (you can't take and get a real angle). See Types of Angles.
Every angle has a supplement.
False for the same reason — an angle must be . A reflex angle like has no supplement since is negative.
Two acute angles can be supplementary.
False — two acute angles are each , so their sum is and can never reach the supplementary total. At most they are complementary or just unrelated.
Two obtuse angles can be complementary.
False — an obtuse angle already exceeds on its own, so two of them blow past instantly. They can't even be supplementary (sum exceeds ).
If two angles are complementary, both must be acute.
True — each is minus a positive angle, so each is strictly between and , which is the definition of acute in Types of Angles.
Complementary and supplementary angles must be adjacent (share a side).
False — the definitions only require the measures to add to or . Two angles in different rooms can be complementary if one is and the other .
A protractor's inner and outer scales give different angles for the same rays.
False — they give the same angle read from opposite directions; the two readings always add to . You must pick the scale whose sits on your baseline ray (see the protractor figure below).
If you measure an angle as on one scale, the other scale reads .
True — the scales run in opposite directions and . This is exactly the "wrong scale" trap: is the supplement of the true reading.
Doubling an angle doubles its complement.
False — the complement of is ; the complement of is , which is not double . Complements shrink, not scale. (E.g. complement ; complement , not .)
Negative angle measures and angles above can appear in measurement.
True in a broader setting — a negative angle means you turned the opposite way (clockwise), and means more than a full turn. But a protractor only reads the plain – (or –) range, so for basic measurement we keep measures in .
Spot the error
"The supplement of is ."
Wrong total — supplements subtract from , not . The correct supplement is ; the is actually the complement.
"These two angles add to , so they're complementary."
Swapped names — adding to makes them supplementary. Complementary is the (corner) sum; supplementary is the (straight line) sum.
"I centered the protractor on the vertex but read from where the ray starts, not where it crosses the curved edge."
You must read where the second ray crosses the protractor's arc, not at the vertex. The vertex is always the center point; the degree value lives out on the semicircular scale (see the protractor figure below).
"My baseline ray sits on the of the right-hand scale, so I'll read the left-hand scale."
Mismatch — you must read the same scale whose your baseline is on. Reading the other scale gives you the supplement ( minus the true angle).
"The angle looks obtuse but my protractor read , so the answer is ."
A visual sanity check flags this: obtuse means (see the type chart), so a reading means you read the wrong scale. The true value is likely .
"A right angle's complement is ."
A right angle's complement is — a valid but degenerate zero angle, which contributes nothing. So people say a right angle has "no useful complement": it already fills the whole by itself.
"Since and its complement add to , and and its supplement add to , the complement plus the supplement equals for any ."
The reasoning double-counts differently but the number is right only as , which is not a constant — it depends on . The statement's fixed "" is the error.
"A turn of and a turn of can't be the same angle because one is negative."
They land the ray in the same place — (clockwise) and (counter-clockwise) differ by one full turn. As positions they coincide; as measures they're written differently.
Why questions
Why does the protractor have two number scales instead of one?
So you can measure from either ray as your baseline without flipping the tool — one scale reads clockwise, the other counter-clockwise, and they always sum to .
Why must the center mark sit exactly on the vertex, not "close enough"?
The angle is defined as the spread from the vertex; even a offset tilts the reference lines and can shift the reading by –. That is precisely why careful centring matters more than it looks — a tiny slip at the pivot fans out into a large error at the arc.
Why is a full turn and not, say, ?
It's a historical Babylonian base- convention: is highly composite (divisible by ), so a circle splits into whole-number parts cleanly. It's a choice of unit, not a law of geometry.
Why are they called "complementary" and "supplementary" — why those words?
From Latin: complementum = "that which completes" a right angle (); supplementum = "something added to complete" a straight angle (). The names encode the target sum.
Why can't a single angle be "its own supplement" unless it's ?
Being your own supplement means , so , giving . Only the right angle splits a straight line into two equal halves.
Why does reading the wrong protractor scale specifically give you " minus the true angle"?
Because the two scales run in opposite directions from the same line, the mark at true angle on one scale lines up with on the other — they partition the semicircle's between them.
Why can't we just eyeball angles instead of using a protractor?
The eye reliably distinguishes broad classes (acute vs obtuse) but not precise degrees; two angles differing by look nearly identical, yet that error propagates badly in later constructions. See Types of Angles for the classes eyeballing can catch.
Why is a reflex angle () still a real angle if it has no supplement?
Having a supplement is a bonus property of angles , not a membership requirement. A reflex angle is a genuine turn — the "long way round" outside a wedge — it just sits past the ceiling where the supplement formula gives negatives.
Edge cases
Is an angle? What is its complement and supplement?
Yes — a angle (rays perfectly overlapping) is a valid but degenerate angle sitting at the lower boundary of the acute range; its complement is and its supplement is .
Is a straight angle () its own supplement's partner — what's its supplement?
Its supplement is . So a straight angle and a zero angle are supplementary, which fits: a full straight line plus "no turn" still spans one straight line.
What is the complement of exactly ?
Exactly — it is the unique self-complementary angle, since . This is the balance point where complement equals the angle.
Can a angle be supplementary to another angle?
Yes — its supplement is . Two right angles are supplementary, which is why a straight line can be split into two equal right angles.
What happens to the supplement as the angle approaches ?
The supplement shrinks toward . In the limit at the supplement collapses to a zero angle — the boundary where a supplement stops being a "real" spread.
Can an angle equal its own supplement and its own complement at the same time?
No — self-supplement forces , self-complement forces , and no single angle is both. The two conditions have different unique solutions.
If two angles are both supplementary and equal, what must each measure?
Each must be , since equal supplements satisfy . Any unequal supplementary pair (like and ) works too, but equal forces the right angle.
What does a reflex angle of have that a "supplement" would deny?
It's a valid angle (the big turn outside a wedge) but has no supplement, since is negative. Its outside partner is the related acute angle, not a supplement.
Is acute, right, or obtuse — and does it have a supplement that's also ?
It is exactly a right angle (neither acute nor obtuse), sitting on the boundary; its supplement is also , so a right angle is supplementary to another right angle. See the type chart above.
Recall One-line self-test
Complement target ::: (the corner / right angle) Supplement target ::: (the straight line) Wrong-scale reading gives you ::: the supplement, true angle Self-complementary angle ::: Self-supplementary angle ::: Degenerate angles ::: (overlap) and (straight line) Reflex angle range ::: between and , has no supplement