Module 19 of 45Lesson 1 of 20 · Unit G01 · Name That Shape
Geometry & Trigonometry · Unit G01 · Foundations
Name That Shape
A gate swings open. A roof meets a wall. A paving slab sits against its neighbour. Every one of those is an angle or a shape with a name, and the name is not decoration — it is what tells you which rule you are allowed to use next. This unit is where the vocabulary of geometry gets nailed down: what to call an angle, how to write it so nobody can misread it, and how to name a triangle, a quadrilateral or a polygon from the marks alone.
The first unit of the Geometry & Trigonometry strand. Nothing here needs any number work beyond adding and dividing whole numbers of degrees — what it needs is precision with words, because in geometry the name of a shape is a claim about its properties.
By the end of this unit you can
- Name any angle from its size: acute, right, obtuse, straight, reflex or a full turn
- Write and read three-letter angle notation, with the vertex in the middle every time
- Name a triangle two ways — by its sides and by its angles — from marks or from measurements
- Tell the six members of the quadrilateral family apart by their properties, not by their look
- Name polygons up to ten sides, and describe exactly what makes one regular
- Read ticks, arcs, right-angle squares and arrowheads, and give the most precise name the marks support
The one idea behind all of it
A shape’s name is a claim about its properties, not a description of how it looks on the page. “Rhombus” claims four equal sides. “Square” claims four equal sides and four right angles. So the way to name anything is to read the marks — the ticks, the arcs, the little square, the arrowheads — and then say the most precise name that everything you can see supports. Turning the page does not change a single one of those marks, which is why a tilted square is still a square.
read the marks → list the claims → give the most precise name
and never claim a property the figure has not marked
Three facts do most of the work in this unit. An angle is named by its size alone, against 90°, 180° and 360°. The middle letter of ∠ABC is the vertex. And regular means equal sides AND equal angles — either one alone is not enough.
Every answer carries its unit: an angle is written with the degree sign. Anything that is not exact is given to 3 significant figures — rounded, never truncated. One interior angle of a regular heptagon is 900° ÷ 7, which is 129° to 3 significant figures.
Where this unit sits
This is the first unit of the Geometry & Trigonometry strand. It builds the vocabulary every later unit leans on: G02 Angle Arithmetic calculates with the angles you learn to name here, G03 Crossing Lines adds the parallel-line properties to the arrowheads, and G05 Cover the Surface needs the quadrilateral family before it can choose an area formula.
No new number work. The arithmetic here is adding and dividing whole numbers of degrees, which Number Unit N01 Know Your Numbers already covers, and ordering values, which N08 The Number Line covers. If a degree total or a division of 360 feels shaky, go back to those two units rather than practising it here.
House rules. Every answer that is a size carries its unit: an angle is written with the degree sign, a length with cm or m. Answers that are not exact are given to 3 significant figures — rounded, never truncated. One interior angle of a regular heptagon is 900° ÷ 7, which is not a whole number of degrees, so it is written 129° to 3 significant figures.
1
Angles, named by size
Acute, right, obtuse, straight, reflex, full turn – and nothing in between
Every angle has a name, and the name comes from one thing only: its size. Six names cover every angle there is, and the boundaries between them are 90°, 180° and 360°. An angle of 89° and an angle of 91° look almost identical and have different names, which is exactly why you compare rather than eyeball.
Every arc is drawn from the two arms it sits between, and every size is measured off the model rather than typed in.
Worked example 1 · Sort the yard
1
A gate post makes an angle of 47° with the fence. Compare it with 90°.
47° is less than 90°
2
Name it.
acute
3
The gate opens until it lies flat against the fence: 180°.
straight angle
4
Swung right round the post and back: 360°.
full turn
5
Three quarters of the way round the post.
270°
6
270° is between 180° and 360°, so name it.
reflex
Six names, and the only thing you need to place any angle among them is where it sits against 90°, 180° and 360°.
| Name | Size | What it looks like | Where you meet it |
|---|---|---|---|
| Acute | more than 0°, less than 90° | sharp, narrower than a corner of a page | the pitch of a roof, a slice of bread |
| Right angle | exactly 90° | marked with a small square, never with an arc | a wall meeting a floor, a corner of a slab |
| Obtuse | more than 90°, less than 180° | open, wider than a corner but not flat | a laptop lid part-way open |
| Straight angle | exactly 180° | a straight line through the vertex | a door swung flat against the wall |
| Reflex | more than 180°, less than 360° | the arc goes the long way round | the outside of a gate’s swing |
| Full turn | exactly 360° | all the way round, back to the start | one complete turn of a wheel |
EXAMPLEA gate turns through three quarters of a full turn. A full turn is 360°, so the gate has turned 3 × 360° ÷ 4 = 270°. That is more than 180° and less than 360°, so it is a reflex angle — and the arc that marks it is drawn the long way round.
Practice · Angles, named by size · 10 questions
P1An angle measures 47°. Name it.Fluency
Answeracute
P2An angle measures 133°. Name it.Fluency
Answerobtuse
P3An angle measures 215°. Name it.Fluency
Answerreflex
P4An angle measures 90°. Name it.Fluency
Answerright angle
P5A door swings from closed until it lies flat against the wall, turning through 180°. Name the angle it has turned through.Applied
Answerstraight angle180° is exactly half a full turn, and the two arms lie in one straight line through the vertex. That is a straight angle — not an obtuse angle, which must be less than 180°.
P6A gate turns through three quarters of a full turn. Calculate the angle and name it.Applied
Answer270°, reflexA full turn is 360°, so three quarters is 3 × 360° ÷ 4 = 270°. That is more than 180° and less than 360°, so it is reflex.
P7Name each of these four angles: 89°, 91°, 180°, 359°.Applied
Answeracute, obtuse, straight angle, reflexThe only comparisons that matter are with 90°, 180° and 360°. 89° is just under 90° so it is acute; 91° is just over, so it is obtuse. Two degrees apart, two different names.
P8Two angles at a point make a full turn together. One of them is 125°. Calculate the other and name it.Applied
Answer235°, reflexAngles at a point add to 360°, so the other is 360° − 125° = 235°. Since 235° is between 180° and 360° it is reflex — the arc goes the long way round.
P9The hands of a clock at 4 o’clock. Calculate the smaller angle between them and name it.Challenge
Answer120°, obtuseThe twelve hour marks share 360° equally, so one hour mark is 360° ÷ 12 = 30°. At 4 o’clock the hands are four marks apart: 4 × 30° = 120°. It is between 90° and 180°, so it is obtuse. The reflex angle on the other side is 240°.
P10Two angles on a straight line: one is five times the other. Determine both, and name each.Challenge
Answer150° (obtuse) and 30° (acute)Angles on a straight line add to 180°. Six equal parts make 180°, so one part is 180° ÷ 6 = 30°. The small angle is 30° and the large one is 5 × 30° = 150°. Check: 30° + 150° = 180°.
2
Notation: the middle letter is the vertex
∠ABC is the angle at B, and the marks on a figure are part of the question
Two rays and a shared end point make an angle, and the shared end point is the vertex. Three-letter notation names the vertex in the middle: ∠ABC is the angle at B, between the arms BA and BC. Everything else on a figure — the ticks, the arcs, the little square, the arrowheads — is part of the question too, and nothing that is not marked may be assumed.
Four rays, two named angles. ∠ABC is the angle at B between BA and BC — not the angle at A.
Worked example 2 · Name and notate
1
Rays BA, BC and BD all leave the point B.
B is the vertex of every angle here
2
Write the angle between BA and BC.
∠ABC
3
Which letter is the vertex?
the middle one, B
4
Write the same angle the other way round.
∠CBA
5
∠ABC = 45° and ∠CBD = 45°, so add them.
∠ABD = 90°
6
Name ∠ABD.
right angle
∠ABC and ∠CBA are the same angle. ∠BAC is a different angle altogether, because the vertex moved to A.
| The mark | What it claims | What it does not claim |
|---|---|---|
| One tick on two sides | those two sides are equal in length | nothing about any other side |
| Two ticks on another pair | that second pair are equal to each other | that they equal the single-ticked pair |
| Matching arcs at two vertices | those two angles are equal in size | that either of them is 90° |
| A small square at a vertex | that angle is exactly 90° | anything about the sides |
| Arrowheads on two sides | those two sides are parallel | that they are equal in length |
EXAMPLEThree rays leave the point B, with C between A and D. If ∠ABC = 62° and ∠CBD = 28°, then ∠ABD = 62° + 28° = 90°, which is a right angle and would be drawn with a small square. Notice that all three names share the middle letter B: one vertex, three angles.
Practice · Notation: the middle letter is the vertex · 10 questions
P11In ∠PQR, state which vertex the angle is at.Fluency
AnswerQ
P12In triangle ABC, write the angle at B using three-letter notation.Fluency
Answer∠ABC
P13State what a small square drawn at a vertex tells you.Fluency
Answerthat angle is exactly 90°
P14State what two matching arcs at two different vertices tell you.Fluency
Answerthose two angles are equal in size
P15In quadrilateral DEFG, write the angle at F in three-letter notation, in both possible orders.Applied
Answer∠EFG and ∠GFEThe vertex letter goes in the middle, and the other two are the neighbouring vertices, in either order. Both names describe the same angle.
P16A student writes “∠ABC = 90°” and marks the square at vertex A. Describe the error.Applied
Answerthe middle letter is the vertex, so ∠ABC is at B, not at AReading the first letter as the vertex is the commonest notation error in the whole strand. Trace A → B → C: the turn happens at B, so the square belongs at B.
P17Triangle ABC has one tick on AB and one tick on BC. Describe what the marks claim, and name the triangle by its sides.Applied
AnswerAB = BC, so the triangle is isoscelesMatching ticks claim equal lengths and nothing else. Two equal sides is the definition of isosceles; the third side is not claimed to be equal to anything, so the triangle is not equilateral.
P18Rays BA, BC and BD leave the point B with C between A and D. ∠ABC = 62° and ∠CBD = 28°. Calculate ∠ABD and name it.Applied
Answer90°, a right angleThe two angles share the vertex B and the arm BC, and C lies between A and D, so they add: 62° + 28° = 90°. An angle of exactly 90° is a right angle, and would be drawn with a small square rather than an arc.
P19Three rays leave B with C between A and D. ∠ABD = 118° and ∠ABC = 47°. Determine ∠CBD and name it.Challenge
Answer71°, acuteBecause C is between A and D, ∠ABC and ∠CBD add to ∠ABD. So ∠CBD = 118° − 47° = 71°, which is less than 90° and so acute. Note that ∠DBC names the same angle.
P20A four-sided shape has two ticks on each of two adjacent sides and one tick on each of the other two. Describe what the marks alone claim, and name the most likely shape.Challenge
Answertwo pairs of equal adjacent sides, so a kiteThe marks claim that one pair of sides that meet at a vertex are equal to each other, and the other pair are equal to each other but not necessarily to the first pair. That is exactly a kite. It could also be a rhombus, but only if the two tick counts matched, which they do not.
3
Triangles, by side and by angle
Every triangle has two names, and the marks tell you both
A triangle always has two names: one from its sides and one from its angles. Equilateral, isosceles and scalene count how many sides are equal. Acute-angled, right-angled and obtuse-angled look at the biggest angle. A triangle can be right-angled and isosceles at the same time, because those two names answer different questions.
Three ticks, two ticks, none: the marks name the triangle by its sides before a single length is measured.
Worked example 3 · Which triangle is which
1
Triangle 1 carries a single tick on all three sides.
all three sides equal
2
Name it by its sides.
equilateral
3
Triangle 2 has sides 5 cm, 5 cm and 8 cm.
exactly two equal
4
Name it by its sides.
isosceles
5
Triangle 3 has angles 90°, 55° and 35°, which add to 180°.
one angle is exactly 90°
6
Name it by its angles, and by its sides.
right-angled, and scalene
Every triangle has both names. Marks give you the side name; angles give you the angle name; a question can ask for either.
| Named by | Name | What it means | How the figure shows it |
|---|---|---|---|
| By its sides | Equilateral | all three sides equal | three single ticks; all three angles are 60° |
| By its sides | Isosceles | exactly two sides equal | two matching ticks; the two angles opposite them are equal |
| By its sides | Scalene | no two sides equal | no ticks match; all three angles are different |
| By its angles | Acute-angled | all three angles less than 90° | no square, no obtuse angle |
| By its angles | Right-angled | one angle exactly 90° | a small square at one vertex |
| By its angles | Obtuse-angled | one angle more than 90° | one angle visibly wider than a corner; there can only be one |
EXAMPLEA triangle has one angle of 90° and two equal sides. The three angles add to 180°, so the other two share 90° between them, and equal sides force equal angles: 45° each. The triangle is right-angled by its angles and isosceles by its sides, and both names are needed to describe it properly.
Practice · Triangles, by side and by angle · 10 questions
P21A triangle has sides 5 cm, 5 cm and 8 cm. Name it by its sides.Fluency
Answerisosceles
P22A triangle has sides 6 cm, 7 cm and 9 cm. Name it by its sides.Fluency
Answerscalene
P23A triangle has sides 4 cm, 4 cm and 4 cm. Name it by its sides.Fluency
Answerequilateral
P24A triangle has angles 90°, 55° and 35°. Name it by its angles.Fluency
Answerright-angled
P25A triangle has angles 40°, 60° and 80°. Name it by its angles, and state the check you used.Applied
Answeracute-angled; the three angles add to 180°40 + 60 + 80 = 180, so the three angles can belong to one triangle. The largest of them is 80°, which is less than 90°, so every angle is acute and the triangle is acute-angled.
P26A triangle has angles 25°, 130° and 25°. Name it by its angles and by its sides.Applied
Answerobtuse-angled, and isosceles130° is more than 90°, so the triangle is obtuse-angled — and no triangle can have two such angles, because two would already pass 180°. The two 25° angles are equal, so the sides opposite them are equal and the triangle is isosceles.
P27A triangle has one angle of 90° and two equal sides. Determine the other two angles and name the triangle two ways.Applied
Answer45° and 45°; right-angled and isoscelesThe angles add to 180°, so the other two share 180° − 90° = 90°. Equal sides mean equal opposite angles, so each is 90° ÷ 2 = 45°. The triangle is right-angled by its angles and isosceles by its sides.
P28Describe whether a triangle can be both equilateral and right-angled.Applied
Answerno: an equilateral triangle has three 60° anglesEquilateral means all three sides equal, which forces all three angles equal. Three equal angles adding to 180° must each be 180° ÷ 3 = 60°, and 60° is not 90°. So no equilateral triangle contains a right angle.
P29Determine whether sides of 3 cm, 4 cm and 8 cm can make a triangle, and describe your reasoning.Challenge
Answerno: 3 + 4 = 7, which is less than 8Two sides have to be able to reach across the third. Here the two short sides together are 3 + 4 = 7 cm, which is shorter than the 8 cm side, so they cannot meet: the shape does not close. This is the triangle inequality, and it has to hold for all three pairings.
P30The angles of a triangle are in the ratio 1 : 2 : 3. Determine the three angles and name the triangle by its angles and by its sides.Challenge
Answer30°, 60°, 90°; right-angled and scaleneThe ratio gives 1 + 2 + 3 = 6 equal parts sharing 180°, so one part is 180° ÷ 6 = 30°. The angles are 30°, 60° and 90°. One of them is exactly 90° so it is right-angled, and because all three angles differ all three sides differ, so it is scalene.
4
The quadrilateral family
Square, rectangle, rhombus, parallelogram, kite, trapezium – and which is which
The six quadrilaterals form a ladder rather than six separate boxes. Going down it, each shape keeps fewer properties: a square has four equal sides and four right angles, a rectangle drops the equal sides, a rhombus drops the right angles, a parallelogram drops both, and a trapezium keeps only one pair of parallel sides. The answer to “name it” is always the most precise name the marks support.
Six four-sided shapes. Nothing but the ticks, the right-angle squares and the arrowheads separates them.
Worked example 4 · The quadrilateral ladder
1
Shape P: four equal sides, four right angles.
square
2
Shape Q: four right angles, adjacent sides 6 cm and 4 cm.
rectangle
3
Shape R: four equal sides, no right angle.
rhombus
4
Shape S: two pairs of parallel sides, sides 5 cm and 3 cm, no right angle.
parallelogram
5
Shape T: two pairs of equal adjacent sides, no parallel pair.
kite
6
Shape U: exactly one pair of parallel sides.
trapezium
Read the ladder downwards and each shape drops one property. Read it upwards and each shape adds one. Every square is a rectangle; no rectangle with unequal sides is a square.
| Name | Sides | Angles | Parallel sides | The test that names it |
|---|---|---|---|---|
| Square | all four equal | all four 90° | two pairs | four equal sides AND four right angles |
| Rectangle | opposite pairs equal | all four 90° | two pairs | four right angles, adjacent sides not equal |
| Rhombus | all four equal | opposite pairs equal | two pairs | four equal sides, no right angle |
| Parallelogram | opposite pairs equal | opposite pairs equal | two pairs | both pairs parallel, no right angle, sides not all equal |
| Kite | two pairs of equal adjacent sides | one pair of opposite angles equal | none | two adjacent pairs equal and no side parallel to another |
| Trapezium | no pattern required | no pattern required | exactly one pair | exactly one pair of parallel sides |
EXAMPLEA quadrilateral has vertices at (0, 0), (4, 0), (5, 3) and (1, 3). Going along the bottom side is 4 right and 0 up, and going along the top side is the same, so that pair is parallel and equal; the other pair matches too. But the sides are 4 units and √10 units, so they are not all equal, and no corner is square. Two pairs parallel and nothing more: a parallelogram.
Practice · The quadrilateral family · 10 questions
P31A quadrilateral has four equal sides and four right angles. Name it.Fluency
Answersquare
P32A quadrilateral has two pairs of parallel sides, no right angle and adjacent sides of different lengths. Name it.Fluency
Answerparallelogram
P33A quadrilateral has exactly one pair of parallel sides. Name it.Fluency
Answertrapezium
P34A quadrilateral has four equal sides and no right angle. Name it.Fluency
Answerrhombus
P35A quadrilateral has four right angles, and two adjacent sides both measure 5 cm. Name it as precisely as you can.Applied
AnswersquareFour right angles alone would make it a rectangle. In a rectangle opposite sides are equal, so if two adjacent sides are also equal then all four are — and four equal sides with four right angles is a square. Say “square”, because it says more than “rectangle”.
P36A quadrilateral has two pairs of equal adjacent sides, one pair of equal opposite angles, and no parallel sides. Name it.Applied
AnswerkiteEqual adjacent sides is what separates a kite from a parallelogram, where the equal sides are opposite each other. No parallel pair rules out the trapezium as well, so kite is the only name left that fits every clue.
P37State whether every square is a rectangle, and whether every rectangle is a square.Applied
Answeryes; noA rectangle is defined by its four right angles, and a square has four right angles, so every square really is a rectangle. The reverse fails as soon as the adjacent sides differ: a 6 cm by 4 cm rectangle has four right angles but not four equal sides.
P38Three angles of a quadrilateral are 85°, 95° and 85°. Determine the fourth and describe what the four angles suggest.Applied
Answer95°; opposite angles are equal, so it could be a parallelogramThe angles of a quadrilateral add to 360°, so the fourth is 360° − (85° + 95° + 85°) = 95°. Two pairs of equal opposite angles is a property of a parallelogram — but only a suggestion, because the angles alone do not tell you the sides.
P39A quadrilateral has vertices A(0, 0), B(4, 0), C(5, 3) and D(1, 3). Name it as precisely as you can and describe how the coordinates prove it.Challenge
AnswerparallelogramGoing along AB is 4 right and 0 up; going along DC is also 4 right and 0 up, so AB and DC are equal and parallel. Going along AD is 1 right and 3 up, the same as BC, so that pair matches too. But AB is 4 units and AD is √10 units, so the sides are not all equal, and the corner at A is not square. Two pairs parallel, nothing more: a parallelogram.
P40A quadrilateral has vertices P(0, 0), Q(3, 1), R(2, 4) and S(−1, 3). Name it as precisely as you can.Challenge
AnswersquareEach side steps 3 one way and 1 the other, so all four sides are the same length, √10. Going along PQ is 3 right and 1 up; going along QR is 1 left and 3 up. Those two turns are at right angles to each other, so the corner at Q is 90°. Four equal sides and a right angle: a square, drawn tilted.
5
Polygons: naming them, and what regular means
Equal sides is half of it; equal angles is the other half
The number of sides gives a polygon its name, and this unit needs them up to ten. Regular is a much stronger word than most students use it as: it means all the sides are equal and all the angles are equal. A rhombus has four equal sides and is not a regular quadrilateral; the square is the regular one.
Both shapes have six sides. Only the left one has six equal sides and six equal angles.
Worked example 5 · Regular or not
1
Hexagon 1: six ticks that match, six arcs that match.
equal sides and equal angles
2
Name it.
a regular hexagon
3
The six interior angles add to 720°, and they are equal.
720° ÷ 6 = 120°
4
Hexagon 2: six ticks that match, no arcs marked.
equal sides only
5
Can you call it regular?
no — the angles are not known
6
What single mark would settle it?
matching arcs at all six vertices
Equal sides alone is not regular. A rhombus has four equal sides and is not a regular quadrilateral; the square is.
| Sides | Name | Angle sum | Interior angle if regular | Exterior angle if regular | Diagonals |
|---|---|---|---|---|---|
| 3 | triangle | 180° | 60° | 120° | 0 |
| 4 | quadrilateral | 360° | 90° | 90° | 2 |
| 5 | pentagon | 540° | 108° | 72° | 5 |
| 6 | hexagon | 720° | 120° | 60° | 9 |
| 7 | heptagon | 900° | 129° | 51.4° | 14 |
| 8 | octagon | 1080° | 135° | 45° | 20 |
| 9 | nonagon | 1260° | 140° | 40° | 27 |
| 10 | decagon | 1440° | 144° | 36° | 35 |
The heptagon and the nonagon are the two rows whose interior angle is not a whole number of degrees, so those two are given to 3 significant figures.
EXAMPLEA regular polygon has an exterior angle of 40°. The exterior angles of any polygon add to one full turn, 360°, and in a regular polygon they are all equal — so there are 360° ÷ 40° = 9 of them. The polygon is a nonagon, and its interior angle is 180° − 40° = 140°.
Practice · Polygons: naming them, and what regular means · 10 questions
P41Name a polygon with 7 sides.Fluency
Answerheptagon
P42Name a polygon with 9 sides.Fluency
Answernonagon
P43State the two conditions a polygon must meet to be called regular.Fluency
Answerall its sides equal and all its angles equal
P44Name a polygon with 10 sides.Fluency
Answerdecagon
P45Determine the sum of the interior angles of a hexagon.Applied
Answer720°A hexagon splits into 6 − 2 = 4 triangles from one vertex, and each triangle contributes 180°: 4 × 180° = 720°. The rule is 180° × (n − 2).
P46Determine one interior angle of a regular octagon.Applied
Answer135°The eight interior angles add to 180° × (8 − 2) = 1080°, and in a regular octagon they are all equal, so each is 1080° ÷ 8 = 135°.
P47Determine one exterior angle of a regular decagon.Applied
Answer36°The exterior angles of any polygon add to one full turn, 360°. A regular decagon has ten equal ones, so each is 360° ÷ 10 = 36°. Check: the interior angle is 180° − 36° = 144°.
P48A hexagon has six equal sides, but two of its angles measure 100°. Describe whether it is regular.Applied
Answerno: regular needs equal angles as well as equal sidesIn a regular hexagon every interior angle is 120°, so an angle of 100° rules it out at once. Equal sides on their own are not enough — the same trap that makes a rhombus not a regular quadrilateral.
P49A regular polygon has an exterior angle of 40°. Determine the number of sides and name it.Challenge
Answer9 sides, a nonagonThe exterior angles add to 360° and they are all equal, so the number of them is 360° ÷ 40° = 9. Nine sides is a nonagon, and its interior angle is 180° − 40° = 140°.
P50One interior angle of a regular polygon is 144°. Determine the number of sides and name it.Challenge
Answer10 sides, a decagonWork with the exterior angle, because those always add to 360°. The exterior angle is 180° − 144° = 36°, so there are 360° ÷ 36° = 10 of them. Ten sides is a decagon.
6
Reading the marks, naming the shape
Ticks, arcs, right-angle squares and arrowheads decide the name – not what it looks like
This is where the unit is really tested. An exam figure gives you marks rather than measurements, and a diagram may be drawn deliberately out of proportion, so measuring it with a ruler answers the wrong question. Read the marks, list what they claim, and then say the most precise name that every claim supports.
Four equal sides on both. The right-angle square on the left is the only difference that matters.
Worked example 6 · Marks first, name second
1
A four-sided shape is drawn leaning over.
leaning tells you nothing
2
It carries a single tick on all four sides.
four equal sides
3
So far the name could be…
rhombus, or square
4
It also carries a small square at one vertex.
that angle is exactly 90°
5
Four equal sides and a right angle.
square
6
Turn the page 30° and re-read the marks.
still a square
The tilted shape in the figure above has vertices at (0, 0), (3, 1), (2, 4) and (−1, 3). All four sides are √10 long and the corner at (0, 0) is a right angle, so it is a square however it is drawn.
What the marks let you claim
- matching ticks → those sides are equal
- matching arcs → those angles are equal
- a small square → exactly 90°
- arrowheads → those sides are parallel
What you may never claim
- “it looks like a right angle” — measure nothing on a figure
- “those sides look equal” — no tick, no claim
- “it is upright, so it is a rectangle”
- “it is tilted, so it is a diamond” — not a mathematical name
EXAMPLEA four-sided shape is drawn leaning over, with one tick on each side and a small square at one corner. The ticks claim four equal sides, which alone would make it a rhombus; the small square claims one right angle, and in a rhombus one right angle forces all four. It is a square, and turning the page would not change a single one of those claims.
Practice · Reading the marks, naming the shape · 10 questions
P51A triangle carries a single tick on each of its three sides. Name it.Fluency
Answerequilateral
P52A quadrilateral carries arrowheads on one pair of opposite sides only. Name it.Fluency
Answertrapezium
P53A quadrilateral carries single arrowheads on one pair of opposite sides and double arrowheads on the other pair, with no right angle marked. Name it.Fluency
Answerparallelogram
P54A triangle carries a small square at one vertex and a single tick on two of its sides. Name it two ways.Fluency
Answerright-angled and isosceles
P55A quadrilateral carries a single tick on all four sides and matching arcs at one pair of opposite vertices, with no right angle marked. Name it.Applied
AnswerrhombusFour matching ticks claim four equal sides. With no right-angle square anywhere you cannot claim 90°, so “square” would be claiming more than the marks support. Four equal sides and no right angle is a rhombus.
P56A quadrilateral carries arrowheads on both pairs of opposite sides and a small square at one vertex, and its adjacent sides are not equal. Name it.Applied
AnswerrectangleBoth pairs parallel makes it at least a parallelogram, and in a parallelogram one right angle forces all four. Adjacent sides that differ rule out the square, so the most precise true name is rectangle.
P57A four-sided shape is drawn leaning over, with a single tick on all four sides and a small square at one vertex. Describe why it is still a square.Applied
Answerturning changes no length and no angle, so it is still a squareThe marks say four equal sides and one right angle, and in a rhombus one right angle forces all four. None of that depends on which way up the shape is drawn, so the name does not change when it leans. “Diamond” is not a mathematical name.
P58Two triangles are drawn: the first has a single tick on all three sides, the second has a single tick on two sides only. Describe the difference the marks make.Applied
Answerthe first is equilateral, the second is only isoscelesThree matching ticks claim all three sides equal, which also forces all three angles to be 60°. Two matching ticks claim only that two sides are equal; the third side is unmarked, so nothing is claimed about it and the triangle is isosceles.
P59A quadrilateral carries arrowheads on one pair of opposite sides, and matching single ticks on the two sides that are not parallel. Name it as precisely as you can.Challenge
Answeran isosceles trapeziumExactly one parallel pair makes it a trapezium. The extra claim is that the two non-parallel sides are equal, and a trapezium with that property is called isosceles — the same word doing the same job as it does for triangles.
P60A pentagon is drawn with a single tick on all five sides and no arcs anywhere. Determine whether you can be sure it is regular, and describe the mark that is missing.Challenge
Answerno; matching arcs at all five vertices are missingFive equal sides is only half of regular. Without any angle marked, the pentagon could be a squashed one whose angles differ, exactly as a rhombus has four equal sides without being a regular quadrilateral. The missing mark is a matching arc at every vertex, which would claim all five angles equal — 108° each.
The naming routine
Four moves, whatever the figure turns out to be.
1
Read the marks before you read the pictureTicks say “these sides are equal”. Arcs say “these angles are equal”. A small square says “exactly 90°”. Arrowheads say “these sides are parallel”. Anything not marked is not known.
2
Count the sides, then look at the anglesThe number of sides gives the family name — triangle, quadrilateral, pentagon, hexagon. The sides and angles together give the exact member of that family.
3
Say the most precise name that is trueA square really is a rectangle and really is a rhombus, but “square” says more, so “square” is the answer. Never claim more than the marks support.
✓
Check by turning itTurn the shape in your head. If the name changes, the name was wrong: no property of a shape depends on which way up it is drawn.
The mistakes that cost the most marks
1Reading ∠ABC as the angle at A
Wrong“∠ABC is the angle at A, because A is written first.” The student marks the wrong corner and every line after it is answering a different question.
RightThe middle letter is the vertex, always. ∠ABC is at B; ∠BAC is at A. If in doubt, trace the path A → B → C: the turn happens at B.
This costs whole questions in the papers, because a wrong vertex is not a small slip — it changes which angle you are talking about. Write the vertex letter down first, then build the three-letter name around it.
2Calling any four-sided shape a rectangle because it looks upright
Wrong“It has four sides and it is standing up straight, so it is a rectangle.” Nothing about being upright says a single angle is 90°.
RightA rectangle claims four right angles. If the figure does not mark them — or state them — you cannot claim them. Four sides with two pairs parallel and no right angle is a parallelogram.
The same trap in reverse marks a parallelogram down when it really is a rectangle. Read the marks: right-angle squares are drawn, never assumed, and being upright is not a property.
3Assuming a shape drawn tilted is no longer a square
Wrong“It is on its corner, so it must be a rhombus, or a diamond.” “Diamond” is not a mathematical name at all.
RightTurning a shape changes no length and no angle. The tilted shape with vertices (0, 0), (3, 1), (2, 4) and (−1, 3) has four sides of √10 and a right angle at (0, 0): it is a square.
Every property that names a shape — equal sides, equal angles, parallel pairs — survives turning it. That is why the check at the end of the routine is “turn it and read the marks again”.
Tasks to submit
Criterion B · Investigating patterns
How many diagonals?
A diagonal joins two corners of a polygon that are not already joined by a side. A triangle has none at all. A quadrilateral has two. After that they build up fast, and the question is how fast, and why. You need nothing but a pencil, a ruler and a careful count — but you do need to be systematic, because the easiest way to get this wrong is to count the same diagonal twice.
A diagonal joins two vertices that are not already joined by a side. Count them here, then predict the octagon.
Part 1 · structured
| Sides | Polygon | Diagonals | Jump from the row above |
|---|---|---|---|
| 3 | triangle | 0 | — |
| 4 | quadrilateral | 2 | 2 |
| 5 | pentagon | ||
| 6 | hexagon | ||
| 7 | heptagon |
- Predict, before you draw anything, whether doubling the number of sides doubles the number of diagonals. Write your prediction down; you will be asked to compare it with what you find.
- Complete the table by drawing each polygon and counting its diagonals for 3, 4, 5, 6 and 7 sides. Draw every diagonal in a different colour from the sides so you can see what you are counting, and work round the vertices in order so that nothing is counted twice.
- Identify the pattern. Look at the jump from each row to the next and describe what those jumps do. Then describe how many diagonals leave a single vertex in each of your polygons, and why that number is not the number of sides.
- Determine a general rule for the number of diagonals of a polygon with n sides. Write it as a formula in n, not in words alone.
- Test and verify your rule on two polygons that are not in the table: one with 8 sides, and one with 12 sides. For the octagon, check your rule by drawing and counting as well — a rule tested only on the numbers it was built from has not been tested.
- Justify your rule. Describe why each vertex contributes exactly n − 3 diagonals, and describe why the total then has to be halved. Say what would go wrong if you forgot the halving.
Part 2 · open
Now you choose what to investigate. Here are three openings — take one, or propose your own. You decide which polygons to try, how many you need, how to record what you find and how you will test it.
- From one corner only. Investigate how many diagonals leave a single vertex, and how many triangles those diagonals cut the polygon into. Determine a rule for each, test both, and describe how your triangle rule connects to the interior-angle sum 180° × (n − 2).
- Diagonals that escape. A polygon does not have to be convex. Investigate how many of the diagonals fall outside a dented polygon — try several five-sided and six-sided shapes with one corner pushed in. Determine whether the total number of diagonals changes, and describe what does change.
- Where they cross. Draw a convex polygon whose diagonals never meet three at a time. Investigate how many crossing points the diagonals make inside it for 4, 5 and 6 sides, predict 7, and determine a rule. Describe the connection between a crossing point and a choice of four corners.
Whichever you choose, take it through the same ladder: predict, identify the pattern, determine a general rule, test and verify it on polygons you have not already drawn, and justify why it must be true.
Submit: your completed table, your prediction written before you drew anything, your rule in n, your check on the octagon and the twelve-sided polygon, and your reasoning — as a scan or a typed document. Marked out of 8 against Criterion B. Completing the table earns the lower bands; the general rule, the test on polygons that are not in the table and the justification are what reach levels 5–8. Part 2 must show your own choice of method.
Remember, you should communicate in an organized and coherent manner.
Criterion D · Applying maths in real-life contexts
Which slabs will pave the patio?
A hardware yard in Gaborone has been asked to quote for paving a patio, and it has sent over its order sheet. The sheet lists five slab shapes, and it tells you a great deal more than you need. Two questions have to be answered before anything is ordered: which of the five shapes will cover the patio with no gaps, and how many slabs the job takes. Neither question is about price, and neither is about how the slabs look.
The order sheet (illustrative figures, written for this task)
- The patio: a rectangle, 4.8 m by 3.6 m
- Slab A: square, 300 mm by 300 mm
- Slab B: rectangle, 600 mm by 300 mm
- Slab C: regular hexagon, 200 mm side
- Slab D: regular pentagon, 200 mm side
- Slab E: equilateral triangle, 300 mm side
- Pallets: 60 slabs to a pallet, whichever shape
- Bedding sand: 4 bags for every square metre of patio
- Yard hours: 07:30 to 17:00, deliveries on Tuesdays and Thursdays
- The yard gate: 2.4 m wide
- Cutting: the yard will cut slabs to order, two working days
These figures are illustrative. The yard, the patio and the slab list are an invented illustrative example written for the task. It is not taken from any business, price list or publication, and no organisation, person or source is named or implied. What is authentic is the shape of the problem: paving really is chosen by whether the slabs fit together with no gaps.
The patio is a rectangle and the slab is a square. Both lengths divide by 0.3 m exactly, which is what makes the count come out whole.
Only 90°, 120° and 60° divide 360° exactly. That is the whole test.
- Identify which entries on the order sheet actually matter for the two questions, and describe, item by item, why the others cannot affect either answer — deal with the yard hours, the bedding sand, the gate width and the cutting service by name. (Strand i — identifying the relevant elements)
- Select the property that decides whether a shape tiles, and describe why it is the right test. Say what has to be true of the angles where several slabs meet at one point, and describe the strategy you will use to count the slabs. (Strand ii — selecting appropriate strategies)
- Apply them. Calculate one interior angle of each of the five slab shapes, and decide for each whether it tiles. Then calculate how many Slab A squares cover the patio, and how many Slab B rectangles would instead. (Strand iii — applying the strategy)
- Describe the accuracy of your answers. The patio is given in metres and the slabs in millimetres, so describe the conversion you had to do and what would have gone wrong without it. Then describe what it tells you that both slab counts come out as whole numbers, and what you would have had to do if they had not. (Strand iv — accuracy of the solution)
- Describe whether your answer makes sense for a real patio. Work out how many full pallets the square slabs need, and describe at least two things the calculation ignores that a paver would not — and say what you would tell the customer to order. (Strand v — does it make sense in context)
Submit: your list of which entries on the sheet matter and which do not, the interior angle of each of the five slab shapes with your verdict for each, both slab counts, the number of pallets, and written answers for strands i, iv and v. Marked out of 8 against Criterion D at Foundations level, where the command term is describe: a solution that gives the numbers and says nothing about them cannot reach the upper bands. State every angle in degrees and every length in metres, and give any answer that is not exact to 3 significant figures.
When you are ready
The mastery quiz is 10 questions drawn at random from a bank of 25. You need 70% to pass and you have three attempts. Type an angle as digits only, with no degree sign, and type a name in lower case. Work through the practice questions first — especially Concept 2, where the middle letter is the vertex, and Concept 6, where the marks decide the name.
Printable version
Download Unit G01 as a PDF — the same lesson, laid out for printing and for working offline.
Download Unit G01 as a PDF — the same lesson, laid out for printing and for working offline.