0580 · 4.1Geometrical terms

Points, lines and angles

0:00A diagram becomes much easier to understand when its labels tell you exactly what to look for. Parallel describes a relationship between lines, while vertex identifies a corner. These words are not just names to memorise. By the end of this video, you will understand how geometrical language describes positions, directions, shapes and their parts. We need only a few starting ideas. Length measures distance, and angles measure turns in degrees.

0:24A complete turn contains three hundred and sixty degrees. Geometry uses ideal shapes, so a drawn line has thickness, but the mathematical line it represents does not. A point is an exact position with no size. We represent it using a small dot, usually labelled with a capital letter. A line is straight and extends indefinitely in both directions. A line segment is the part between two endpoints, while a plane is a flat surface extending indefinitely.

0:51A vertex is a corner where sides or edges meet. The plural is vertices, so a triangle has three vertices. An angle forms where two rays meet at a common endpoint. That endpoint is its vertex, and the angle measures the turn between the rays. When an angle is named using three letters, the middle letter identifies its vertex. In angle PQR, the turn happens at Q. A right angle is exactly ninety degrees.

1:15Four right angles together make one complete turn. An acute angle is greater than zero degrees but less than ninety degrees. An obtuse angle is greater than ninety degrees but less than one hundred and eighty degrees. A reflex angle is greater than one hundred and eighty degrees but less than three hundred and sixty degrees. Two rays that do not form a straight line make a smaller angle and a reflex one together.

1:35In the diagram, the acute angle at Q is forty degrees, so the reflex one is three hundred and twenty degrees. An interior angle lies inside a polygon at one of its vertices. An exterior angle is formed between a side and the extension of an adjacent side. It is not simply any angle drawn outside the shape. Following the sides around a polygon makes exterior angles useful for describing changes in direction.

Parallel, perpendicular and bearings

1:59Parallel lines lie in the same plane and never meet, however far they are extended. Their perpendicular separation stays constant. Matching arrow marks show which lines or sides are parallel. These marks describe a relationship, rather than the direction in which you should travel. Perpendicular lines meet at a right angle. A small square at their intersection usually marks that ninety-degree angle. A perpendicular bisector combines two conditions: it meets a segment at a right angle and divides it into equal halves.

2:26Being perpendicular alone does not guarantee that the segment has been bisected. For our first worked example, the segment from P to Q is eight centimetres long. The line through M and R divides it into two equal parts. 8 ÷ 2 gives four centimetres for each part. The angle at M is ninety degrees, so that line is the perpendicular bisector of the segment from P to Q. This answer needs both pieces of evidence.

2:49Equal halves establish the bisector condition, while ninety degrees establishes the perpendicular condition. A line passing through the midpoint at a different angle would not be a perpendicular bisector. We will study how to draw one accurately in the video on geometrical constructions. A bearing describes direction using an angle measured clockwise from north. The north line is drawn at the point you measure from. Bearings are written using three figures, so an angle of twenty-five degrees needs a leading zero.

3:16In the diagram, the bearing of Q from P is zero two five degrees. We will explore bearing diagrams and calculations further in the video on scale drawings.

Polygons, similarity and congruence

3:25A polygon is a closed, flat shape made entirely from straight sides. A triangle has three sides, and a quadrilateral has four. Names within these groups describe extra properties, such as equal lengths or right angles. Judge a shape by these properties, not by whether it looks upright or tilted. An equilateral triangle has three equal sides and three equal angles. Each angle is sixty degrees. An isosceles triangle has two equal sides, and the angles opposite those sides are equal.

3:52A scalene triangle has no equal sides, so its three angles are also different. A right-angled triangle contains one angle of ninety degrees. This classification describes an angle, whereas isosceles and scalene describe side lengths. A triangle can therefore be both right-angled and isosceles. Rotating that triangle does not change either description because its lengths and angles remain unchanged. A rectangle has four right angles, with opposite sides equal and parallel.

4:20A square also has four right angles, but all four sides are equal. A rhombus has four equal sides, with opposite sides parallel. Its angles need not be right angles, so a rhombus does not have to be a square. A square has every property of both, so it counts as a rectangle and as a rhombus. A parallelogram has two pairs of parallel opposite sides. Its opposite sides are equal, and its opposite angles are equal.

4:44Rectangles, rhombuses and squares satisfy that definition too. These names can overlap because a special shape retains the properties of the broader group it belongs to. A kite has two pairs of equal adjacent sides. Adjacent means next to each other, so each equal pair shares a vertex. In this kite, the shorter equal sides meet at P, while the longer pair meets at R. Do not confuse adjacent sides with opposite sides, which lie across the shape from each other.

5:10A trapezium has exactly one pair of parallel sides. In this trapezium, the top and bottom sides are parallel. The two slanting sides are not parallel, and they would meet if extended. Cambridge uses the word trapezium in this sense, so look for that one pair of arrow marks. A pentagon has five sides, a hexagon has six, an octagon has eight, and a decagon has ten. A regular polygon has all sides equal and all interior angles equal.

5:36The hexagon shown is regular, and each of its interior angles is one hundred and twenty degrees. An irregular polygon fails at least one of those conditions. Equal sides alone are insufficient, as a rhombus without right angles demonstrates. Congruent shapes have exactly the same shape and size. One may be moved, rotated or reflected and still fit exactly over the other. Similar shapes have the same shape, but their sizes may differ.

6:00Corresponding angles are equal, and corresponding lengths have the same scale factor. Corresponding means matching the same position or role in each shape. The scale factor compares a length in the image with its corresponding original length. Every original length is multiplied by this factor. A factor greater than one enlarges the shape, while a positive factor below one makes it smaller. Our second example compares a rectangle measuring six centimetres by four centimetres with one measuring twelve centimetres by eight centimetres.

6:28Comparing the widths gives 12 ÷ 6, which equals two. The heights confirm the same factor because 8 ÷ 4 also equals two. Their corresponding angles are ninety degrees, so the rectangles are similar but not congruent. The direction of the comparison matters. From the larger rectangle back to the smaller one, the scale factor is 1/2. Congruence would require equal corresponding lengths, not just equal corresponding angles. For this syllabus point, you need to interpret congruence, but you are not expected to prove that two shapes are congruent.

Solids and nets

7:00A solid is a three-dimensional shape. A face is a flat part of its boundary, while surface is the broader term that includes curved parts. An edge is a boundary where faces meet, or where a flat face meets a curved surface. A vertex is a corner where edges meet. A cube has six square faces, all the same size. A cuboid has six rectangular faces, with opposite faces matching.

7:22A prism has a constant cross-section along its length. For example, a triangular prism has matching triangular ends joined by rectangular side faces. A cylinder has two parallel circular faces joined by a curved surface. A pyramid has a polygonal base and triangular faces meeting at one vertex. A cone has a circular base and a curved surface narrowing to a vertex. The pyramid’s side faces are flat, but the cone’s side surface is curved.

7:49A sphere has a completely curved surface, with every surface point the same distance from its centre. A hemisphere is half a sphere, with a circular flat face. A frustum is the portion remaining when the top of a cone or pyramid is removed. The cut is parallel to the base. A net is a flat arrangement that folds to form the surface of a solid. The cube net shown contains six squares, one for each face.

8:12Their arrangement matters because they must meet without overlapping when folded. Recognising nets is part of this vocabulary, while drawing them belongs in geometrical constructions.

Circle terms and diagram clues

8:22A circle consists of points in a plane at a fixed distance from its centre. A radius joins the centre to the circumference, which is the circle’s boundary. The plural of radius is radii. A diameter joins two points on the circumference through the centre, so its length is twice the radius. A chord is a straight segment joining any two points on the circumference. A diameter is therefore a special chord, but most chords do not pass through the centre.

8:46A semicircle is half a circle, formed by dividing the circle along a diameter. Its boundary includes a semicircular arc and the diameter itself. An arc is part of the circumference between two points. Usually, those points divide the boundary into a shorter minor arc and a longer major arc. If the endpoints form a diameter, both arcs are semicircles of equal length. A tangent is a straight line touching the circle at just one point.

9:11A sector is a region bounded by two radii and an arc. It resembles a slice cut from a circular pizza because its straight boundaries reach the centre. A segment is bounded by a chord and an arc instead. The difference is in the boundaries, not simply whether the shaded region looks large or small. For our third worked example, we name three features of this diagram. The straight path from P to Q joins two points on the circumference, so it is a chord.

9:36The shading is bounded by that chord and the shorter arc, so it is a minor segment. The line at the bottom touches the circle only at T, so it is a tangent. An examination question may ask you to name a shape, identify a marked feature, or interpret a description. Start with the given properties rather than guessing from appearance. A tilted square remains a square, and an inaccurate sketch does not override stated lengths.

9:58Matching ticks show equal lengths, while matching arrows show parallel sides. These identification methods must work without a calculator on Paper two, and they also apply on Paper four. Our calculations here give exact answers, so rounding is unnecessary. Common mistakes include calling every chord a diameter and confusing a segment with a sector. Avoid them by checking whether the line passes through the centre and which boundaries enclose the region.

10:23Geometrical vocabulary tells you what an object is and how its parts relate. Equal, parallel and perpendicular describe different properties, while similar and congruent compare whole shapes. For circles, identify endpoints, the centre and the boundaries of any shaded region. Understanding those relationships turns a crowded diagram into information you can use.