From scissors and clock hands to the pure idea: two rays pinned at a single point O, and the shaded opening between them — the angle ∠AOB.
Turn from length to turning. So far a straight figure was something you could lay along a ruler and read off its length. But pin two rays at one point and swing one of them open, and you create something new: a gap, a spread, an opening. That opening is an angle. And just like a segment has one number attached to it — its length — an angle has a number too: how far you turned to open it. This lesson meets the angle, names it, and learns to measure it in degrees with a protractor.
13.5.1 What an angle is, and how to name it
Take a single point and shoot two rays out of it. The shared starting point is the vertex; the two rays are the sides of the angle. The angle itself is the opening between the sides — the amount of turn you'd make to swing one side onto the other.
The angle ∠AOB: vertex O, sides ray OA and ray OB, and the shaded opening between them.
There are three common ways to name this angle, and the symbol for "angle" is ∠:
Three letters — ∠AOB or ∠BOA. The two outer letters name a point on each side; the middle letter is always the vertex. So the vertex O sits in the middle, every time.
One letter — ∠O — naming the angle just by its vertex, but only when there is no chance of confusion.
A number — ∠1 — a label you write inside the opening, handy in a busy figure.
Key idea
In a three-letter angle name, the vertex letter goes in the middle. ∠AOB and ∠BOA are the same angle (vertex O); ∠OAB is a different angle (vertex A).
Watch out
You may not shorten an angle to ∠O if several angles share the vertex O. If both ∠AOB and ∠BOC live at O, then "∠O" is ambiguous — nobody can tell which opening you mean. Use the full three letters.
13.5.2 Measuring with degrees
How wide is an angle? We need a unit of turn. Imagine swinging a ray all the way around until it lands back where it started — one full turn. Slice that full turn into 360 equal wedges. Each tiny wedge is one degree, written 1°. So a full turn is 360°, and an angle's measure just counts how many of those degree-wedges fit inside its opening.
The tool that lays those degrees against a real angle is the protractor: a half-disk with the 0°–180° marks already printed on its rim. To measure, put the protractor's center on the vertex, line one side up with 0°, and read where the other side crosses the scale.
Try it Swing the ray and read the protractor
Drag the slider to open the angle. Watch the degree count climb — and notice how the name of the angle changes as it grows.
Angle
Reading tip
A good protractor has two rows of numbers, one running 0→180 and one running 180→0. Always start your count from the 0 on the side you lined up, so you follow a single row all the way around. Crossing rows is the classic way to misread a protractor.
13.5.3 Special angles you should know on sight
A few openings come up so often they each earn a name. Sort every angle by where its measure θ falls between 0° and 360°:
The family of named angles — acute, right (marked with a small square), obtuse, straight, and the full turn.
Name
Measure θ
Looks like
Acute
0° < θ < 90°
a sharp opening
Right
θ = 90°
a square corner ⊾
Obtuse
90° < θ < 180°
a wide opening
Straight
θ = 180°
a flat line
Full
θ = 360°
one whole turn
The right angle is the anchor of the whole family: exactly a quarter turn, the corner of every sheet of paper, marked not with an arc but with a small square ⊾. Anything below it is acute ("a-cute little angle"); anything between a right angle and a straight line is obtuse. A straight angle is a half-turn — the two sides point in exactly opposite directions and form a straight line — and a full angle brings you all the way home, 360°.
Key idea
The slider in 13.5.2 already announces each of these as you pass it: under 90° it reads acute, exactly 90° it reads right and draws the square, between 90° and 180° it reads obtuse, and at 180° it reads straight. Scroll back and find each one.
13.5.4 Minutes and seconds — the 60-carry
Whole degrees are not always fine enough. Just like an hour splits into minutes and seconds, a degree splits into 60 minutes, and a minute splits into 60 seconds:
1° = 60′ and 1′ = 60″
The mark ′ means minutes of arc and ″ means seconds of arc — same names as clock time, because they share the same base-60 idea. And that means they carry the same way: whenever minutes reach 60, they become 1° and you carry; whenever seconds reach 60, they become 1′ and you carry.
Worked example
Add 38′ + 35′. That's 73′ — but 73 is more than 60, so 60 of those minutes become a whole degree: 73′ = 60′ + 13′ = 1° 13′. The 60 carried, exactly like adding 38 minutes to 35 minutes on a clock.
Try it Add two amounts of minutes and watch the 60 carry
Set two minute counts. When their total reaches 60′, one whole degree carries out — see the result rewrite itself as D° M′.
First40
Second35
Watch out
Minutes and seconds do not roll over at 100 — they roll over at 60. So 75′ is not "0.75°"; it is 60′ + 15′ = 1° 15′. Treat every degree-measure like a little clock.
★ Recap
An angle is the opening between two rays sharing one endpoint — the vertex. The rays are the sides.
Name it ∠AOB with the vertex letter in the middle, or ∠O / ∠1 when there's no confusion.
Measure turning in degrees: a full turn is 360°, so 1° is one of 360 equal wedges. A protractor reads it off.
Sort by size: acute (0°–90°), right (90°), obtuse (90°–180°), straight (180°), full (360°).
Finer units carry by 60: 1° = 60′ and 1′ = 60″.
✎ Exercises 13.5
An angle has vertex O and sides through A and B. Write its name two correct ways, and explain why ∠OAB would be wrong.
Answer
∠AOB or ∠BOA — the vertex O must sit in the middle. ∠OAB puts A in the middle, so it names an angle with vertex A, a different angle entirely.
Classify each angle as acute, right, obtuse, or straight: 45°, 90°, 130°, 180°.
Answer
45° is acute; 90° is right; 130° is obtuse; 180° is straight.
Convert 1.5° into degrees and minutes.
Answer
The 0.5° is half a degree, and half of 60′ is 30′. So 1.5° = 1° 30′.
How many degrees are in a straight angle? How many in a full turn?
Answer
A straight angle is a half-turn, 180°. A full turn is 360° (twice as much).
Express 2° 15′ entirely in minutes.
Answer
Each degree is 60′, so 2° = 120′. Then 120′ + 15′ = 135′.
Add 38′ + 35′ and write the result in degrees and minutes.
Answer
38′ + 35′ = 73′. Since 73 ≥ 60, carry one degree: 73′ = 60′ + 13′ = 1° 13′.
🎯 Quick check
Six questions to lock it in. Tap the answer you think is right.
§ For teachers and parents
The big idea
An angle measures turn, not length. A child who has just mastered measuring segments naturally — and wrongly — expects "longer sides" to mean "bigger angle." Keep returning to the opening: it is the spread between the rays, and the drawn sides could be any length at all. The protractor widget above makes this concrete because the rays stay the same length while only the opening changes.
The second classic stumble is the protractor double scale. Because most protractors print both a 0→180 and a 180→0 row, students read 50° as 130° (or vice versa). The fix is a habit: line one side up on a zero, then count along the single row that started there. The reasonableness check — "this looks acute, so the answer should be under 90°" — saves more points than any rule.
The third is treating minutes and seconds as decimals. 75′ is not 0.75°; it is 1° 15′, because the units carry at 60, exactly like clock time. The carry widget makes that base-60 rollover visible. Conversions both directions (1.5° → 1° 30′, and 2° 15′ → 135′) are worth practicing until automatic.
Common Core alignment: 4.MD.C.5 (an angle as a turn through fractions of a circle, measured in one-degree wedges), 4.MD.C.6 (measure and sketch angles with a protractor), and 4.MD.C.7 (angle measure is additive — the foundation for the addition and bisecting work in 13.6).