C4.1; E4.1 - Lines, angles and polygons

Points, lines and angle types

Can a square become a different shape just by standing on one corner? Its appearance changes, but its lengths and angles do not. To recognise it reliably, start with the language that describes those properties.

A point marks an exact position; the dot we draw makes that position visible. A vertex is a corner where sides meet; the plural is vertices. A straight line continues indefinitely in both directions. A line segment has two endpoints, and a ray has one endpoint and continues in one direction. A triangle has three straight sides and three vertices; a rectangle has four of each.

An angle measures a turn between two rays with the same endpoint. A full turn is $360^\circ$; a straight angle is $180^\circ$. In $\angle ABC$, the middle letter $B$ is the vertex, and the arms run from $B$ towards $A$ and $C$. A marked arc tells you which turn is intended when the same arms enclose two possible angles.

The strict boundaries matter: $90^\circ$ is right, not acute or obtuse. An interior angle lies inside a polygon at a vertex. An exterior angle is the turn between a side and the extension of its neighbouring side outside the polygon. For a convex polygon, use the outside angle adjacent to the interior angle, as marked below. Angle calculations come later; first identify the correct angle.

[DIAGRAM: asset_name: Lines and angle language; asset_slug: lines_and_angle_language; caption: Line markings and angle arcs describe properties regardless of orientation.; description: Three panels show parallel sloping lines and a perpendicular pair, acute right obtuse and reflex angle examples with correctly sized arcs, and a triangle with one side extended and its interior and adjacent exterior angles labelled.]

Use the boundaries in the table to choose between similar-looking descriptions.

Triangles and quadrilaterals

Classify a shape using given measurements and markings, rather than how it happens to look. A shape can belong to more than one family.

An equilateral triangle has three equal sides and three equal angles, each $60^\circ$. An isosceles triangle has at least two equal sides; the angles opposite those equal sides are equal. An equilateral triangle satisfies this condition too, but equilateral is its more specific name. A scalene triangle has no equal sides and no equal angles. A right-angled triangle has one $90^\circ$ angle. The last description is about angles, so a right-angled triangle can also be isosceles or scalene. For example, sides meeting at a right angle may both be $5\,\mathrm{cm}$, giving an isosceles right-angled triangle.

A quadrilateral is a polygon with four sides. Here are the properties that distinguish the special quadrilaterals. A diagonal joins two non-neighbouring vertices; bisect means cut into two equal parts.

For a kite, adjacent means sharing a vertex: it is not the same as opposite. For a general trapezium, do not assume equal sloping sides or equal base angles. In the usual classification used here, a trapezium has exactly one pair of parallel sides; when a problem supplies markings, describe those markings explicitly.

[DIAGRAM: asset_name: Special quadrilaterals; asset_slug: special_quadrilaterals; caption: Tick marks, parallel arrows and right-angle squares show the defining properties.; description: Six separately labelled exact quadrilaterals show a square rectangle rhombus parallelogram kite and trapezium. Equal-side ticks and parallel-side arrows distinguish adjacent from opposite equal sides, and right-angle squares appear only on the square and rectangle.]