GCSE · Maths · AQA · Spec 8300

Geometric terms and notation

Turn a triangle upside down, stretch it, flip it. ∠ABC is still the same angle, because in geometry a name is an address, not a description of the picture.

Triangle ABC

Move the triangle. The names stay put.
72°58°50°14.012.411.2ABC

∠BAC 72°. ∠ABC 58°. ∠ACB 50°. side 𝑎 · BC 14.0. side 𝑏 · CA 12.4. side 𝑐 · AB 11.2. Classification: The names follow the vertices. Relationship: ∠ABC is the angle at B, the middle letter (∠CBA names the same angle). Side a is BC, the side opposite A; side b is CA, opposite B; side c is AB, opposite C.

∠BAC72°∠ABC58°∠ACB50°side 𝑎 · BC14.0side 𝑏 · CA12.4side 𝑐 · AB11.2Drag A, B or C

The names follow the vertices

∠ABC is the angle at B, the middle letter (∠CBA names the same angle). Side a is BC, the side opposite A; side b is CA, opposite B; side c is AB, opposite C.

Drag A or C left and right, or drag B up and down. Each time you move one vertex, watch the three side readouts: exactly one of them never changes. It is always the side the moving vertex does not touch, and it carries that vertex's letter in lower case.

Watch out: ∠ABC is not the angle at A. Read the middle letter: ∠ABC is the angle at B. Side a is not a side that touches A. It is BC, the side opposite A.

The words behind the letters

Say the definition to yourself before you turn each card.

Parallel and perpendicular

Marks, not looks

What does each diagram actually prove about the pair of lines? Sort by the marks and statements, not by how the lines look.

Still to sort

Shown parallel (0)

Matching arrowheads, or the symbol ∥

Where the line is: The arrowheads must match: one > with one >, or >> with >>.

Shown perpendicular (0)

A small square where they meet, a stated 90°, or the symbol ⊥

Where the line is: The square is drawn in the corner where the two lines meet; it means the angle there is exactly 90°.

Not shown either way (0)

No mark and no statement, so you cannot say

Where the line is: Looking parallel or looking square is not evidence. Without a mark or a statement, it is not known.

8 of 8 still to sort.

Parallel lines stay the same distance apart and never meet. Perpendicular lines meet at a right angle (90°).

Watch out: Arrowheads pair up only when they match. A line marked > is parallel to the other lines marked >, not to a line marked >>.

Polygons and symmetry

Regular, symmetrical, or both?

A polygon is a closed flat shape with straight sides. It is regular when all its sides are equal and all its angles are equal. It has line (reflection) symmetry if a mirror line splits it into two halves that match exactly. It has rotation symmetry if a turn of less than a full turn about its centre fits it exactly onto itself; the order is how many times it fits in one full turn. Tick every set each shape belongs to.

  • A Regular
  • B Line symmetry
  • C Rotation symmetry (order 2 or more)
  1. Equilateral triangle
  2. Square
  3. Regular hexagon
  4. Rectangle (not a square)
  5. Rhombus (not a square)
  6. Parallelogram (not a rectangle or a rhombus)
  7. Isosceles triangle (not equilateral)
  8. Kite (not a rhombus)
  9. Scalene triangle

From words to a diagram

Sketch it from the description

Sketch triangle PQR from this description: ∠PQR = 90°. Side r = 5 cm and side p = 8 cm. Point S lies on PR, point T lies on QR, and ST ∥ PQ.

  1. Sketch a triangle and label its vertices P, Q and R with capital letters.
  2. missing step
Which line is step 2?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

A few letters and marks say exactly which point, side or angle you mean, however the figure is drawn.

What you need to know

  • A point is named with a capital letter, and a line segment by its two end points, such as AB.
  • An angle is named with three letters, and the middle letter is the vertex: ∠ABC (or ∠CBA) is the angle at B.
  • In triangle ABC, the side opposite vertex A is labelled a, the side opposite B is b, and the side opposite C is c.
  • Matching arrowheads show parallel lines (∥); a small square shows a right angle, so the lines meeting there are perpendicular (⊥).
  • A regular polygon has all sides equal and all angles equal. A polygon may have line symmetry, rotation symmetry, both or neither.

The big picture

Geometry has its own written language. Points are named with capital letters, a line segment by its two end points, and an angle by three letters with the vertex in the middle, so ∠ABC is the angle at B. In triangle ABC, side a is opposite vertex A, side b is opposite B and side c is opposite C. On a diagram, matching arrowheads show parallel lines and a small square shows a right angle; without a mark or a statement, you cannot assume either. A polygon is regular when all its sides and all its angles are equal, and polygons can have line symmetry, rotation symmetry, both or neither.

Key points

1To find an angle from its name, go straight to the middle letter: that is the corner where the angle sits.
2To find side a, look for the one side that does not touch A.
3Never read parallel or perpendicular from how a diagram looks. Only a mark or a written statement shows it.
4Pair arrowheads by matching them: > goes with >, and >> goes with >>.
5A regular polygon with n sides has n lines of symmetry and rotation symmetry of order n, but symmetry alone does not make a shape regular.
6When drawing from a description, give every phrase its own mark or label on the sketch.

Worked example

Problem

Sketch quadrilateral ABCD from this description: AB ∥ DC, AD ⊥ DC, AB = 4 cm and DC = 7 cm.

⚠ Watch out

Reading an angle's name from its first letter. ∠XYZ is the angle at Y, not at X, because the vertex is always the middle letter. The same slip happens with sides: side x is YZ, the side opposite X, not a side that runs into X.

🧠

Memory hook

Middle for the corner, opposite for the side, marks not looks.

✓

Check yourself

A sketch of EFGH has one arrowhead on EF, one on HG, and a small square at H. What do these marks say in notation? Check: EF ∥ HG and ∠EHG = 90°.

Flashcards

(12)
How is a point named?
With a capital letter, such as A. A point is an exact position with no size.
What is the difference between a line and a line segment?
A line goes on forever in both directions. A line segment is the part between two end points, named by them, such as AB.
What is a vertex?
A corner where two or more sides or edges meet. The plural is vertices.
What is an edge of a solid?
A line segment where two faces meet. A cube has 12 edges.
What is a plane?
A perfectly flat surface that extends forever in every direction.
Which angle does ∠PQR name?
The angle at Q. The middle letter of a three-letter angle name is always the vertex.
In triangle PQR, which side is labelled q?
PR, the side opposite vertex Q (the one side that does not touch Q).
What do parallel lines look like on a diagram, and what symbol is used?
They carry matching arrowheads (> with >). In writing, AB ∥ CD means AB is parallel to CD.
How is a right angle shown on a diagram, and what does ⊥ mean?
By a small square in the corner. AB ⊥ CD means AB is perpendicular to CD: they meet at 90°.
What is a polygon?
A closed flat shape made of straight sides.
What makes a polygon regular?
All its sides are equal AND all its angles are equal. A rhombus fails on angles; a rectangle fails on sides.
What is the order of rotation symmetry?
The number of times a shape fits exactly onto itself in one full turn. Order 1 means no rotation symmetry.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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