Types of angles — acute, right, obtuse, straight, reflex, complete: Worked examples

This page is the "no scenario left behind" workbench for the six angle types. Before we solve anything, we lay out every kind of question this topic can ask. Then we work one example for each cell — so when a new problem appears, you already know which drawer it belongs to.

If any word here feels new (right angle, reflex, boundary), it is fully rebuilt below from a single picture.

The scenario matrix

Every angle problem in this topic is really asking: "where does this number sit on the rotation line from $0°$ to $360°$ ?" Look at the rotation line first — it is the spine of everything.

Figure — Types of angles — acute, right, obtuse, straight, reflex, complete

Here is the full matrix of case-classes. Every example below is tagged with the cell it covers.

Cell	Case class	What makes it tricky	Example
A	Interior value (acute / obtuse / reflex)	Pick the right open interval	Ex 1, Ex 2
B	Exact boundary ( $90°, 180°, 270°, 360°$ )	Is it "in" a region or "on" a fence?	Ex 3
C	Zero / degenerate ( $0°$ )	Two rays lie on top of each other	Ex 4
D	Reflex ↔ non-reflex conversion	The "long way round"	Ex 5
E	Limiting behaviour (just under a post, e.g. $89.999°$ )	What the type is approaching	Ex 6
F	Real-world word problem (clock / turn)	Translate a story into a measure	Ex 7
G	Exam twist (find the unknown angle)	Work backwards from a relationship	Ex 8
H	Sum / chain of angles around a point	All pieces must total $360°$	Ex 9

Definition The boundary posts, in plain words

$== 0° ==$ : the two rays sit exactly on top of each other — no opening at all.
$== 90° ==$ (right): a perfect square corner, a quarter turn.
$== 180° ==$ (straight): the two rays point in exactly opposite directions — a flat line.
$== 270° ==$ (three-quarter, sometimes "reflex-right"): three-quarters of a turn — this is the reflex partner of $90°$ . It sits inside the reflex region, so an angle of exactly $270°$ is genuinely a reflex angle; it is only special because it is a clean landmark, not because it earns a brand-new one-word class like "right" or "straight".
$== 360° ==$ (complete): a full spin, back to the start.

An angle is acute if $0° < θ < 90°$ , right at $90°$ , obtuse if $90° < θ < 180°$ , straight at $180°$ , reflex if $180° < θ < 360°$ , and complete at $360°$ .

Why intervals with $<$ and not $\leq$ ? The strict signs $<$ (read "less than", no equals) mean the boundary values are not inside the region. $90°$ is not acute and not obtuse — it has its own name, "right". Fences belong to nobody; they are their own thing (Cell B). Keep that in mind — it is the single most common trap. The one exception is $270°$ : it is a landmark post but it lives strictly between $180°$ and $360°$ , so it still counts as reflex — Example 3 shows exactly why.

The worked examples

Cell A — interior values

Cell B — exact boundary

Worked example Example 3 — sitting ON a fence-post (all four)

Classify each of $θ = 90°$ , $θ = 180°$ , $θ = 270°$ , $θ = 360°$ .

Forecast: Are these acute? obtuse? reflex? (Trick question — and watch $270°$ carefully.)

Notice each value equals a landmark post. $90°$ , $180°$ , $270°$ , and $360°$ are all landmark values, not chosen at random inside a region.
- Why this step? Three of them ( $90°, 180°, 360°$ ) are region-boundary fences, so the strict inequalities ( $<$ , never $\leq$ ) mean no region contains them. We must handle each landmark deliberately.
Give the three true fences their own one-word names. $90° =$ right, $180° =$ straight, $360° =$ complete.
- Why this step? The matrix reserved special single-value names precisely for the posts that separate regions.
Handle $270°$ separately — it is NOT a separating fence. Test it against the reflex region: is $180° < 270° < 360°$ ? Yes ⇒ $270°$ is reflex.
- Why this step? Unlike $90°, 180°, 360°$ , the value $270°$ does not sit between two named regions — it sits strictly inside the single reflex region. So it gets no brand-new one-word class; it is simply a reflex angle (a clean "three-quarter turn" landmark, the reflex partner of $90°$ ).

Answers: $90°$ right, $180°$ straight, $270°$ reflex, $360°$ complete.

Verify: $90, 180, 360$ each fail every strict two-sided region test (so each is a named separating fence); $270$ passes $180 < 270 < 360$ , so $270°$ is reflex. ✔

Cell C — zero / degenerate

Worked example Example 4 — the collapsed angle

Two rays start from the same vertex and point in the same direction. What is the angle, and what type?

Forecast: Is there even an angle here?

Measure the opening between the rays. They overlap perfectly, so the rotation from one to the other is $0°$ .
- Why this step? An angle is the amount of turning needed to swing one ray onto the other. Zero turning ⇒ zero angle.
Classify $0°$ . Our acute rule needs $0° < θ$ , strictly. Since $θ = 0°$ is not greater than $0°$ , it is not acute. It is the zero (degenerate) angle — the lowest fence-post itself.
- Why this step? Same fence-post logic as Cell B: the boundary $0°$ is its own case, not part of the acute region.

Answer: $0°$ , a zero / degenerate angle — not acute.

Verify: $0 < 0$ is false, so $0°$ fails the acute test and is a boundary value. ✔

Cell D — reflex ↔ non-reflex conversion

Worked example Example 5 — the two angles between two rays

Two rays make a $75°$ opening the "short way". Find the reflex angle (the long way round) and classify it.

Forecast: Bigger or smaller than $180°$ ?

See both angles at once. Around the shared vertex, the short opening ( $75°$ ) and the long opening together sweep the whole circle.
- Why this step? A full turn around a point is always $360°$ ; the two openings between two rays are just the two pieces of that full turn.
Subtract from a complete angle. $θ_{r} = 360° - 75° = 285°$ .
- Why this step? If the two pieces add to $360°$ , the long piece is whatever is left after removing the short piece. This is exactly the "angles round a point sum to 360°" idea used elsewhere.
Classify $285°$ . Is $180° < 285° < 360°$ ? Yes ⇒ reflex.
- Why this step? We must confirm the name of the answer, not just its size. Trapping $285°$ between the straight post ( $180°$ ) and the complete post ( $360°$ ) proves it lands in the reflex region — which matches the "long way round" we were asked for.

Answer: reflex, $285°$ .

Verify: $75° + 285° = 360°$ (the two pieces rebuild the full turn), and $180 < 285 < 360$ ⇒ reflex. ✔

Cell E — limiting behaviour

Worked example Example 6 — creeping up to a fence

An angle grows: $89°$ , then $89.9°$ , then $89.99°$ , ... What type is each, and what is it approaching?

Forecast: Do any of these become right angles?

Classify each value. Every one satisfies $0° < θ < 90°$ (since each is strictly below $90°$ ), so every one is acute.
- Why this step? No matter how close we crawl to $90°$ , as long as we have not reached it, the strict inequality $θ < 90°$ still holds — the number is still trapped in the acute region.
Describe the limit. As the value approaches $90°$ it looks visually almost like a right angle, but it stays acute until it equals $90°$ exactly.
- Why this step? This is the whole point of strict fences: the type flips only at the post, never just before it. $89.999°$ is acute; $90.000°$ is right.

Answer: all acute; the type only becomes "right" the instant it hits $90°$ .

Verify: $89.99 < 90$ is true ⇒ acute; the boundary flip happens exactly at equality, not before. ✔

Cell F — real-world word problem

Worked example Example 7 — clock hands at 2 o'clock

At 2:00, what is the smaller angle between the hour and minute hands, and its reflex?

Forecast: Acute or obtuse for the small one?

Find the angle of one hour-step. The clock face is a complete angle, $360°$ , split into $12$ equal hour marks. Each step is $\frac{360°}{12} = 30°$ .
- Why this step? Twelve equal slices must add to the full turn, so each slice is the full turn divided by $12$ .
Count the steps. At 2:00 the minute hand is on $12$ and the hour hand is on $2$ : that is $2$ steps apart. Small angle $= 2 \times 30° = 60°$ .
- Why this step? Two equal slices stacked give twice one slice.
Classify the small angle. Is $0° < 60° < 90°$ ? Yes ⇒ acute.
- Why this step? A measure alone is not an answer to "what type"; we must trap $60°$ between the zero post and the right-angle post to earn the name "acute".
Reflex (long way). $360° - 60° = 300°$ , and $180° < 300° < 360°$ ⇒ reflex.
- Why this step? The two hands, like any two rays, cut the clock into a short piece and a long piece summing to $360°$ (Cell D logic reused).

Answer: small angle $60°$ (acute); reflex $300°$ .

Verify: $\frac{360}{12} = 30$ , $2 \times 30 = 60$ , and $60 + 300 = 360$ . ✔

Cell G — exam twist (work backwards)

Worked example Example 8 — find the unknown from a relationship

Angle $α$ is supplementary to a $118°$ angle. Classify $α$ .

Forecast: Will $α$ be acute or obtuse?

Recall the supplementary relationship. Supplementary angles add to a straight angle, $180°$ : $α + 118° = 180°$ .
- Why this step? The word "supplementary" is a shortcut for "sums to $180°$ " — the same $180°$ post from our matrix.
Solve for $α$ . $α = 180° - 118° = 62°$ .
- Why this step? Subtracting the known piece from the total leaves the unknown piece.
Classify. Is $0° < 62° < 90°$ ? Yes ⇒ acute.
- Why this step? The question asks for a type, not just a number; trapping $62°$ between the zero post and the right-angle post proves the name is "acute".

Answer: $α = 62°$ , acute.

Verify: $62° + 118° = 180°$ (supplementary holds) and $62 < 90$ ⇒ acute. ✔

Cell H — angles chained around a point

Worked example Example 9 — filling the full turn

Three angles meet at one point with no gaps: $x$ , $2 x$ , and a right angle. Find $x$ and classify each of the three.

Forecast: Will $2 x$ be acute, right, or obtuse?

Write the "around a point" equation. All angles filling a full turn add to a complete angle: $x + 2 x + 90° = 360°$ .
- Why this step? No gaps and no overlaps means the pieces exactly reassemble the $360°$ complete angle.
Combine and solve. $3 x + 90° = 360° \Rightarrow 3 x = 270° \Rightarrow x = 90°$ .
- Why this step? Gather like terms, undo the $+ 90°$ , then undo the $\times 3$ .
Classify the three angles. $x = 90°$ ⇒ right; $2 x = 180°$ ⇒ straight; the given $90°$ ⇒ right.
- Why this step? Feed each solved value back through the fence-post test — $90°$ and $180°$ are exact boundaries (Cell B).

Answer: $x = 90°$ (right), $2 x = 180°$ (straight), third angle right.

Verify: $90° + 180° + 90° = 360°$ — the three pieces rebuild the full turn. ✔

Recall Quick self-test

Every problem in this topic reduces to one move — what is it? Answer ::: Trap the number between two fence-posts ( $0, 90, 180, 270, 360$ ) and read the region's name; if it sits on a separating fence, it gets that fence's special name — except $270°$ , which sits inside the reflex region.

Why is $89.99°$ not a right angle? Answer ::: Because the acute region uses a strict $<$ , so the type only flips to "right" at exactly $90°$ — never just before.

A short angle is $110°$ ; what is its reflex and type? Answer ::: $360° - 110° = 250°$ , which is reflex (since $180 < 250 < 360$ ).