Geometrical language and plane shapes
Describe lines, angles, triangles and quadrilaterals using properties rather than appearances.
Section Paper 3 · Core · Calculator
Core and Extended. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Compare lengths and recognise a right angle.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What angle do perpendicular lines make?
What you will learn
- Use precise geometrical terms for lines and angles.
- Classify triangles, quadrilaterals and polygons by their properties.
- Distinguish congruence from similarity.
Name the objects and relationships
A point marks a position; a vertex is a corner where edges or sides meet. A line extends in both directions, while a line segment has two endpoints. A plane is a flat surface extending without a boundary. Parallel lines in one plane never meet; perpendicular lines meet at 90°. A perpendicular bisector crosses a segment at its midpoint at a right angle.
An acute angle is between 0° and 90°, a right angle is 90°, an obtuse angle is between 90° and 180°, and a reflex angle is between 180° and 360°. Interior angles lie inside a polygon; exterior angles are formed by a side and the extension of an adjacent side. In angle ABC, the middle letter B names the vertex.
Classify by properties, not orientation
An equilateral triangle has three equal sides, an isosceles triangle has at least two equal sides, and a scalene triangle has no equal sides. A right-angled triangle has one 90° angle; these descriptions can overlap, as in a right-angled isosceles triangle. Rotating a shape does not change its classification.
A parallelogram has two pairs of parallel opposite sides. A rectangle is a parallelogram with four right angles; a rhombus has four equal sides; a square has both properties. A kite has two pairs of equal adjacent sides. A trapezium has a pair of parallel sides. A pentagon, hexagon, octagon and decagon have 5, 6, 8 and 10 sides. A regular polygon has equal sides AND equal angles, whereas an irregular polygon fails at least one condition.
Same shape and same size
Congruent shapes have the same shape and size, even when turned, reflected or moved. Similar shapes have equal corresponding angles and one common ratio between corresponding lengths. That ratio is the scale factor; scale factor one gives congruent shapes. Match vertices before deciding which sides correspond.
A sketch can suggest a property but cannot prove it. Use stated lengths, angle measures and standard markings: equal-length ticks, parallel arrows and right-angle squares. In this syllabus you need the language of congruence; formal triangle congruence proofs are not required. A shape drawn like a square is not necessarily a square without supporting information.
Pause and explain
A polygon has equal sides but unequal angles. Is it regular?
Worked example
A quadrilateral has four equal sides and an angle of 70°. Is it a square, a rhombus or neither?
Show the worked solution
- Four equal sides give the defining side property of a rhombus.
- A square would require every angle to be 90°; the stated 70° rules that out.
- Its orientation on the page does not affect either conclusion.
Answer A rhombus, but not a square.
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Classify a triangle with side lengths 5 cm, 5 cm and 8 cm. What can you conclude about its base angles?
Give me a hint
Use the equal sides rather than guessing from a sketch.
Compare my reasoning
- Two sides are equal, so the triangle is isosceles.
- The angles opposite the two equal sides are equal.
- No angle size is given; do not claim a right angle.
Isosceles; the two base angles are equal.
Look for these in your work
- Used the stated side lengths.
- Did not assume an unstated angle measure.
Describe the difference between a segment's midpoint and its perpendicular bisector.
Give me a hint
One is a point and one is a line.
Compare my reasoning
- The midpoint divides the segment into two equal lengths.
- The perpendicular bisector passes through that midpoint.
- It is a line meeting the segment at 90°, not the midpoint itself.
Midpoint: a point; perpendicular bisector: the right-angle line through it.
Look for these in your work
- Distinguished point from line.
- Included both bisection and perpendicularity.
A designer enlarges a regular hexagon by scale factor 2. Is the copy similar, congruent and regular?
Give me a hint
Track lengths and angles separately.
Compare my reasoning
- All corresponding lengths double, while angles stay equal.
- The copy is similar but has a different size, so it is not congruent.
- Its six equal sides and six equal angles remain, so it is still regular.
Similar and regular; not congruent.
Look for these in your work
- Applied one common length factor.
- Distinguished size from shape.
Common mistakes
- Treating a rotated square as a different type of shape.
- Calling equal-sided polygons regular without checking angles.
- Using congruent when only the shape, not the size, matches.
What can you explain now?
Which letter is the vertex of angle PQR?
Compare with the explanation
Q
The middle letter names the point where the two arms of the angle meet.
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Make it stick
Tomorrow, sketch a non-square rhombus and a right-angled isosceles triangle. Label only properties you can justify.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.