8.01a–8.01c, 8.01e, 8.01f - Geometrical language and measurement

Points, lines and planes

A carpenter is told, “Put a bar through A and B, then put another bar perpendicular to it.” What information does that give, and what information is still missing?

A point gives an exact position and has no size. A dot marks it on a drawing; a capital letter names it. A line segment has two endpoints. Segment AB runs from A to B and has a finite length. A line through A and B continues without end in both directions; the short piece of ink on the page represents that whole line. A ray has one endpoint and continues indefinitely in one direction.

[DIAGRAM: asset_name: Lines and a plane; asset_slug: lines-and-plane; caption: Endpoints, arrows and a right-angle square communicate different facts.; description: Three panels show a finite segment AB, two parallel lines l and m crossed by a perpendicular line n, and a cuboid with a labelled vertex and edge, standing in a pale plane extending beyond its base. Parallel arrows and a right-angle square identify the stated relationships.]

The face of a tabletop models part of a plane; the plane itself does not stop at the table's edge. In three dimensions, two lines which do not meet need not be parallel: they might lie in different planes. Here, a flat geometry diagram represents one plane unless the question says otherwise.

A vertex is a corner where sides or edges meet; the plural is vertices. In a solid, an edge is a line where two faces meet. A polygon's straight boundary pieces are usually called its sides. Capital letters can name their endpoints: side AB joins vertices A and B.

One arrow on each of two lines marks them as parallel. Matching short ticks on segments mark equal lengths. Different numbers of ticks distinguish different equal-length groups. A small square marks a right angle. These marks are mathematical information; a line merely looking horizontal, equal or perpendicular is not enough.

Choose the statement which uses the information correctly.

Naming and labelling angles

An angle measures the turn between two rays with a common endpoint, called its vertex. One complete turn is $360^\circ$; a quarter-turn is $90^\circ$.

[DIAGRAM: asset_name: Angle names and triangle labels; asset_slug: angle-names-and-labels; caption: The middle letter names an angle's vertex. A side's lower-case letter sits opposite its matching capital.; description: Four small ray diagrams show acute, right, obtuse and reflex angles, with the measured turns highlighted. A separate triangle ABC has side a on BC, side b on AC and side c on AB, and an arc at B.]

The boundaries matter: $90^\circ$ is right, not acute or obtuse, and $180^\circ$ is straight, not reflex. Two rays can describe a smaller angle and the reflex turn the other way. An arc or the wording tells you which is intended.

In $\angle ABC$, B is the vertex: the angle is between BA and BC. Reversing the outside letters gives the same smaller angle, $\angle CBA$. Do not move the middle letter. You may write “angle B” if only one angle at B is possible; three letters remove ambiguity when several rays meet there.

In triangle ABC, the standard convention puts side $a$ opposite angle A, so $a=BC$. Similarly, $b=AC$ and $c=AB$. Upper-case letters name vertices and their angles; lower-case letters name opposite side lengths. A length and an angle are different quantities even when they share a letter's name.

Read the letters before imagining the picture's orientation.

Naming polygons

A shape's name describes its properties, even when the shape is tilted or stretched on the page.

A circle is not a polygon because its boundary is curved. An open chain of four segments is not a quadrilateral because it is not closed. Equal sides alone do not make a polygon regular: a sloping rhombus has equal sides but unequal angles.

[DIAGRAM: asset_name: Naming polygon families; asset_slug: polygon-families; caption: Count sides, then use equality and parallel marks to identify special families.; description: A labelled gallery shows scalene, isosceles and equilateral triangles, a square, rectangle, parallelogram, rhombus, kite and trapezium, and a pentagon, hexagon and octagon. Equal-side ticks, parallel arrows and right-angle squares show the defining information rather than relying on orientation.]

A triangle has three sides. A scalene triangle has three different side lengths; an isosceles triangle has at least two equal sides; an equilateral triangle has three equal sides. Thus an equilateral triangle has the equal-side property of an isosceles triangle too. A right-angled triangle has a right angle: this describes angles rather than side equality, so a triangle can be both right-angled and isosceles.

A quadrilateral has four sides. Learn these descriptions:

A square also satisfies the rectangle, rhombus and parallelogram properties. Names can overlap; “rectangle” does not mean “a shape which cannot be a square”. The properties lesson will examine their angles and diagonals in detail.

For larger polygons, a pentagon has five sides, a hexagon has six and an octagon has eight. These names count sides; they do not imply regularity. For example, an irregular pentagon still has five sides.

Check both requirements rather than choosing by appearance.

Using geometric instruments

Use each instrument for the information it controls: a ruler for length, a protractor for angle and compasses for a fixed distance from a centre.

To measure a segment, place the ruler's zero mark at one endpoint and read the other endpoint, looking straight down. The physical end of a ruler is not necessarily zero. If zero is damaged or inconvenient, subtract the two readings: endpoints at $2.3\,\mathrm{cm}$ and $7.1\,\mathrm{cm}$ are $7.1-2.3=4.8\,\mathrm{cm}$ apart. To construct a specified segment, mark its endpoints against the scale, then join them with the ruler.

[DIAGRAM: asset_name: Ruler protractor and compass methods; asset_slug: geometric-instruments; caption: Align the zero, the protractor centre and the compass point before making a mark.; description: Three panels show a ruler with segment endpoints at 1.5 cm and 6.7 cm, a semicircular protractor centred at O with baseline OA and rays at 68 degrees and 112 degrees on the scale starting at zero on the right, and a circle with centre C, a radius CP, and a fixed compass opening.]