FG1, FG4, HG1, HG4 - Describing shapes and symmetry

Geometric notation

If someone says “draw a diamond”, can you tell whether they mean a square or a rhombus with no right angles? A useful description must identify properties, rather than how a shape happens to sit on the page.

A point is a position, labelled with a capital letter such as $A$. A straight line extends indefinitely in both directions. The line segment $AB$ is the part between its endpoints $A$ and $B$; $AB$ can also denote its length when the context is a measurement. A plane is a flat surface extending in every direction. A sheet of paper represents part of a plane.

Neither relationship depends on the page: parallel lines can slope, and perpendicular lines need not be horizontal and vertical. Matching arrow marks show parallel lines; a small square marks a right angle. Matching short ticks show equal lengths. Arrow marks and length ticks mean different things.

[DIAGRAM: asset_name: Triangle labels and line markings; asset_slug: geometric-notation; caption: Capital letters name vertices. The middle letter names an angle's vertex.; description: Triangle ABC has vertex A at the top, B lower left and C lower right. Lower-case a labels BC, b labels AC and c labels AB. Angle ABC is highlighted at B. Separate sloping lines carry matching parallel arrows, and another sloping pair meets with a right-angle square.]

A vertex is a corner; the plural is vertices. A polygon has straight sides meeting at vertices. For a solid, an edge is where two faces meet; the solid's faces can lie in different planes. A triangle has three sides and three vertices.

Name a triangle by its vertices, for example triangle $ABC$. In an angle name, the middle letter is the vertex: $\angle ABC$ is the angle between $BA$ and $BC$, at $B$. The reversed name $\angle CBA$ refers to the same angle. A single letter, such as angle $B$, is enough only when it is unambiguous. The usual lower-case labels $a,b,c$ name the sides opposite $A,B,C$, respectively; they are not extra vertices.

Changing the vertex changes the opposite side, even though the triangle stays the same.

Use the written information and markings before trusting a sketch's appearance.

Try the notation with different letters.

Triangles and quadrilaterals

Describe a shape by facts that remain true when it is moved, turned or reflected.

A polygon is a closed plane figure whose boundary consists of straight line segments. A triangle has three sides; a quadrilateral has four. A circle is not a polygon because its boundary is curved. The ordinary polygons considered here have no crossing sides. A polygon can be concave, with a corner pointing inward, or convex, with every corner pointing outward.

These descriptions answer different questions. A triangle can be both right-angled and isosceles. Under the “at least two” definition, an equilateral triangle is also a special isosceles triangle. An acute-angled triangle has three angles below $90^\circ$; an obtuse-angled triangle has one angle between $90^\circ$ and $180^\circ$.

Why do equal sides give equal opposite angles? Fold an isosceles triangle along the line from the vertex between the equal sides to the midpoint of its base. The equal sides fit onto each other and the base ends swap, so the two base angles fit too. With three equal sides, the same reasoning applies at every vertex. The triangle angle sum, developed in the polygon-angles lesson, then gives $60^\circ$ each.

[DIAGRAM: asset_name: Special triangles and quadrilaterals; asset_slug: shape-properties; caption: Equal-length ticks, parallel arrows and right-angle squares record properties, regardless of orientation.; description: An equilateral triangle, a right-angled isosceles triangle and a scalene triangle appear above examples of a square, rectangle, parallelogram, rhombus, kite and trapezium. Matching ticks distinguish equal side sets, arrows identify parallel opposite sides and small squares mark right angles. The rectangle is not a square, the rhombus has no right angles, the kite has two unequal pairs of adjacent equal sides and the trapezium has exactly one parallel pair.]

The word adjacent means next to each other, sharing a vertex. Opposite sides do not share a vertex. A diagonal joins two non-adjacent vertices. Use these distinctions in the quadrilateral definitions.

Here a trapezium means a quadrilateral with exactly one parallel pair. Some definitions allow “at least one”; when a question gives a property explicitly, follow that wording. A kite drawn with two distinct side lengths has just one symmetry line. A rhombus also satisfies the adjacent-equal-pair condition, but has additional symmetry.

Bisect means divide into two equal parts. The diagonals of a parallelogram bisect each other; they do not necessarily meet at right angles. You can see the bisection by a half-turn about their crossing: opposite vertices exchange places, so each half of a diagonal matches the other half. For a rectangle, folding along either line through the midpoints of opposite sides also shows its opposite sides are equal. For a rhombus or kite, folding along the appropriate diagonal swaps the two matching sides, so it halves the other diagonal at right angles. Later geometric proofs turn these observations into formal arguments.