Angles

Study Sheet

Angles

Measuring and classifying turns

Naming Angles

Concept
The Big Idea
40°acuteright130°obtuse
Less than, exactly, and more than a quarter turn.

An angle measures a turn. A quarter turn is 90∘90^\circ (a right angle), and a half turn is 180∘180^\circ (a straight angle).

An angle measures how far one ray has turned away from another. A full turn is 360∘360^\circ, so a quarter turn — the corner of a square — is 90∘90^\circ, a right angle. Angles smaller than a right angle are acute; larger ones (but less than a straight line) are obtuse.

Tip
The three names

Less than 90∘90^\circ is acute. Exactly 90∘90^\circ is right. Between 90∘90^\circ and 180∘180^\circ is obtuse.

Example
This angle
55°
A 55° turn: less than the square corner of a right angle.

It opens less than a quarter turn, and it measures 55∘55^\circ, so it is acute.

Example
Reading the right scale: 130∘130^\circ
01803015060120909012060150301800
The outer numbers count from the right-hand 0; the inner, fainter ones from the left.

Put the protractor's centre on the vertex and line up 00 with one ray. This ray starts at the 00 on the right, so read the scale that counts up from that 00: the other ray crosses 130130. The other scale shows 5050 at the same spot; it belongs to angles measured from the left. The angle opens wider than a right angle, so 130∘130^\circ (obtuse) makes sense and 50∘50^\circ does not.

Side note
Where it goes wrong

Measuring from the wrong zero on the protractor. Every protractor has two scales; use the one that starts at 00 on the ray you began with, and sanity-check: does the angle look smaller or bigger than a right angle?

Try it
Try it: name the angles

Name each angle: 90∘90^\circ, 145∘145^\circ, 12∘12^\circ, 180∘180^\circ.

Show the answer

90∘90^\circ is right. 145∘145^\circ is between 90∘90^\circ and 180∘180^\circ: obtuse. 12∘12^\circ is less than 90∘90^\circ: acute. 180∘180^\circ is a half turn: a straight angle.

In Plain Terms & More Examples

Concept
In plain terms

An angle measures how far something turns. Compare it to a right angle (90∘90^\circ): smaller is acute, bigger (up to 180∘180^\circ) is obtuse.

Example
A 120∘120^\circ angle
120°
More than a right angle, less than a straight line.

It is more than 90∘90^\circ but less than 180∘180^\circ, so it is obtuse.

Example
A 30∘30^\circ angle
30°
A third of a right angle.

It is less than 90∘90^\circ, so it is acute.

Side note
Where it goes wrong
40°40°
Short rays or long rays: both angles are 40°.

Judging an angle by the length of its sides. Longer rays do not make a bigger angle. The angle measures the turn between the rays, and it stays the same however far the rays are drawn.

Try it
Try it: pizza slices

A round pizza is cut into 88 equal slices from the centre. What angle does one slice make at the centre? Is it acute?

Show the answer

The 88 slices share the full turn, 360∘360^\circ: 360÷8=45∘360 \div 8 = \mathbf{45^\circ}. That is less than 90∘90^\circ, so it is acute.

Adding and Splitting Angles

Concept
Angles add
25°60°85° in all
25° and 60° side by side make 85°.

When a ray splits an angle into two parts, the parts add to the whole. On a straight line the parts add to 180∘180^\circ.

Example
The missing piece
35°?
A right angle (square corner) split into 35° and the missing piece.

A right angle is split into 35∘35^\circ and another angle. The other is 90∘−35∘=55∘90^\circ - 35^\circ = 55^\circ.

Example
Three angles on a line
50°70°?
The three angles together fill the straight line: 180°.

Three angles sit side by side on a straight line: 50∘50^\circ, 70∘70^\circ and an unknown one. Together they make a half turn, 180∘180^\circ. So the unknown angle is 180∘−50∘−70∘=60∘180^\circ - 50^\circ - 70^\circ = 60^\circ.

Side note
Where it goes wrong

Using the wrong total. Angles that fill a right angle add to 90∘90^\circ; angles along a straight line add to 180∘180^\circ; angles all the way round a point add to 360∘360^\circ. Look for the square corner mark or the straight line before you subtract.

Try it
Try it: on a line

Two angles make a straight line. One is 112∘112^\circ. What is the other, and is it acute or obtuse?

Show the answer

180∘−112∘=68∘180^\circ - 112^\circ = \mathbf{68^\circ}, which is less than 90∘90^\circ: acute.

Word Problems

Concept
A clock is a circle of turns

A full turn is 360∘360^\circ. A clock face splits it into 1212 equal hour-spaces, so each hour-space is 360∘÷12=30∘360^\circ \div 12 = 30^\circ. Count the hour-spaces between the hands and multiply by 30∘30^\circ.

Every turning question, whether it is about clock hands, a door or a skateboarder, comes back to the full turn of 360∘360^\circ. Splitting the full turn into equal parts is division. That is one reason 360360 is so handy: it divides evenly by 2,3,4,5,6,8,102, 3, 4, 5, 6, 8, 10 and 1212.

Example
The clock
121234567891011

At 33 o'clock the hands make a quarter turn: 14\tfrac14 of 360∘=90∘360^\circ = 90^\circ, a right angle.

Example
Four o'clock
123456789101112
Four hour-spaces of 30° between the hands.

At 44 o'clock the hour hand points at 44 and the minute hand at 1212. That is 44 hour-spaces: 4×30∘=120∘4 \times 30^\circ = 120^\circ, an obtuse angle.

Side note
Where it goes wrong

Counting minute marks as degrees. Between 1212 and 44 there are 2020 minute marks, but each mark is 360∘÷60=6∘360^\circ \div 60 = 6^\circ, not 1∘1^\circ. 20×6∘=120∘20 \times 6^\circ = 120^\circ, the same as 4×30∘4 \times 30^\circ.

Try it
Try it: turning

A skateboarder does a "540540". How many full turns is that, and how much more?

Show the answer

540=360+180540 = 360 + 180: one full turn and a half turn — she ends up facing backwards.

Perimeter and Area on Grid Paper

Concept
Around versus inside

Perimeter is the distance around a shape: add the lengths of all its sides. Area is the number of unit squares that cover it; for a rectangle it is length×width\text{length} \times \text{width}. Perimeter is measured in units, area in square units.

A rectangle 66 long and 44 wide is 44 rows of 66 squares, so it holds 4×64 \times 6 squares. That is the whole reason area is length times width. Perimeter is different: an ant walking round the edge covers 6+4+6+46 + 4 + 6 + 4 units and never goes inside.

Example
A 66 by 44 rectangle
64
4 rows of 6 unit squares.

Perimeter: 6+4+6+4=206 + 4 + 6 + 4 = 20 units (or 2×6+2×4=202 \times 6 + 2 \times 4 = 20). Area: 44 rows of 66 squares, 6×4=246 \times 4 = 24 square units.

Example
An L-shape
324375
Six sides to add for the perimeter; two rectangles for the area.

Split it into two rectangles: a 77 by 33 piece along the bottom and a 33 by 22 piece on top. Area: 7×3+3×2=21+6=277 \times 3 + 3 \times 2 = 21 + 6 = 27 square units. For the perimeter, walk round and add every side: 7+3+4+2+3+5=247 + 3 + 4 + 2 + 3 + 5 = 24 units.

Example
Finding a missing side

A rectangle has area 3636 square units and width 44 units. Its length times 44 is 3636, so the length is 36÷4=936 \div 4 = 9 units. Now the perimeter: 9+4+9+4=269 + 4 + 9 + 4 = 26 units.

Side note
Where it goes wrong

Mixing up the two measures, or their units. Writing "6×4=246 \times 4 = 24 units" for the perimeter of the 66 by 44 rectangle is wrong twice: perimeter adds the sides (2020 units), and 2424 is the area, measured in square units.

Try it
Try it: the garden

A garden is 88 m long and 33 m wide. How much fence goes around it, and how much ground does it cover?

Show the answer

Fence (perimeter): 8+3+8+3=228 + 3 + 8 + 3 = \mathbf{22} m. Ground (area): 8×3=248 \times 3 = \mathbf{24} square metres (m2^2).

Why It Works

Why 360360? A circle can be split into any number of equal turns, and 360360 was chosen because it divides so evenly: by 2,3,4,5,6,8,9,10,122, 3, 4, 5, 6, 8, 9, 10, 12 and more. That is why halves, thirds, quarters and sixths of a turn are all whole numbers of degrees. Angles add like lengths: two angles that share a ray combine into one whose measure is the sum, which is the fact behind "the angles on a straight line add to 180∘180^\circ" — half a turn.

Tip
Reading a proof

Do not read a proof the way you read a rule. Cover the last line, ask what the previous line forces, and only then uncover it. If a step feels like magic, that is the step to work by hand with a concrete number.