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4b · E4.4–E4.6

Similarity, symmetry and angles

Topic 4 Geometry · Core and Extended · Papers 1–4

🎯What you need to be able to do

  • Calculate angles using angles at a point, on a straight line, vertically opposite, and in triangles and quadrilaterals — and give the reasons.
  • Use corresponding, alternate and co-interior angles with parallel lines, naming the property.
  • Use interior and exterior angles of regular and irregular polygons.
  • Recognise line symmetry and the order of rotational symmetry; for Extended, planes and axes of symmetry of solids EXTENDED.
  • Calculate lengths in similar shapes.
  • Use the area and volume scale factors of similar shapes and solids, and show that two triangles are similar EXTENDED.

📚The mathematics

Angle facts and reasons

angles at a point: \( 360^{\circ} \)
on a straight line: \( 180^{\circ} \)
vertically opposite angles are equal
triangle: \( 180^{\circ} \); quadrilateral: \( 360^{\circ} \)

With a pair of parallel lines cut by a transversal: corresponding angles are equal (F-shape), alternate angles are equal (Z-shape), and co-interior angles add up to \( 180^{\circ} \) (C-shape). When a question says “give a reason”, write the property in words; the letters F, Z and C are not accepted as reasons.

Two parallel horizontal lines cut by a slanted transversal. At the upper crossing the marked angle is 68 degrees. At the lower crossing, the corresponding angle is 68 degrees, the alternate angle is 68 degrees, and the co-interior angle is 112 degrees.
From one angle of \( 68^{\circ} \): corresponding and alternate angles are also \( 68^{\circ} \); the co-interior angle is \( 180^{\circ} - 68^{\circ} = 112^{\circ} \).

Polygons

sum of interior angles \( = (n - 2) \times 180^{\circ} \)
exterior angles sum to \( 360^{\circ} \)
regular: each exterior angle \( = \dfrac{360^{\circ}}{n} \)
interior \( + \) exterior \( = 180^{\circ} \)

For a regular polygon, work with the exterior angle: it is the fastest route to \(n\), and to the interior angle.

Symmetry

A line of symmetry is a mirror line; the order of rotational symmetry is how many times the shape looks the same in one full turn (every shape has at least order 1). A regular \(n\)-gon has \(n\) lines and order \(n\). A parallelogram has no lines of symmetry but order 2. EXTENDED Solids have planes of symmetry (a cuboid with three different dimensions has 3) and axes of rotational symmetry (a cone or cylinder has one axis of infinite order).

Similar shapes

Similar shapes have equal angles and all lengths multiplied by the same scale factor \(k\). Match corresponding sides carefully — redraw the shapes the same way round if necessary. EXTENDED Areas scale by \( k^{2} \) and volumes by \( k^{3} \):

length factor \(k\)
area factor \( k^{2} \) (including surface area)
volume factor \( k^{3} \) (including capacity and mass of the same material)
Two similar cuboids. The small one is 1 by 1 by 2 units; the large one has every length doubled, 2 by 2 by 4. Beneath them: length factor 2, area factor 4, volume factor 8.
Doubling every length multiplies areas by \( 2^{2} = 4 \) and volumes by \( 2^{3} = 8 \).

EXTENDED To show two triangles are similar, show two pairs of equal angles, giving a geometric reason for each (for example, “corresponding angles, DE parallel to BC” and “common angle”).

✏️Worked example

(a) (Non-calculator.) Each exterior angle of a regular polygon is \( 24^{\circ} \). Find the number of sides and the size of each interior angle. [2] (b) (Non-calculator.) In the diagram of parallel lines above, find the co-interior angle to \( 68^{\circ} \), giving a reason. [2] (c) (Calculator.) EXTENDED Two cones are similar, with heights 8 cm and 12 cm. The smaller has surface area 96 cm², and the larger has volume 540 cm³. Find the surface area of the larger and the volume of the smaller. [4]

(a) \( n = 360 \div 24 = 15 \) sides; interior angle \( = 180 - 24 = 156^{\circ} \).

(b) \( 180 - 68 = 112^{\circ} \), because co-interior angles between parallel lines add up to \( 180^{\circ} \).

(c) Length factor \( k = \tfrac{12}{8} = 1.5 \). Area factor \( 1.5^{2} = 2.25 \): larger surface area \( = 96 \times 2.25 = 216 \) cm². Volume factor \( 1.5^{3} = 3.375 \): smaller volume \( = 540 \div 3.375 = 160 \) cm³.

Check it. In (c), going from large to small must make both numbers smaller, and going from small to large bigger: 96 → 216 and 540 → 160 ✓. In (a), \( 15 \times 156 = 2340 = 13 \times 180 \), the interior sum ✓.
Using the length factor for volumes. \( 540 \div 1.5 = 360 \) is the most common wrong answer in (c). Square for areas, cube for volumes.

📝Practise

Questions in the style of the current papers. EXTENDED marks Extended-only content.

1. (Non-calculator.) An isosceles triangle has one angle of \( 40^{\circ} \) between its two equal sides. Find the other two angles. [2]
\( (180 - 40) \div 2 = 70^{\circ} \) each (base angles of an isosceles triangle are equal).
2. (Non-calculator.) Find the size of each interior angle of a regular 12-sided polygon. [2]
Exterior \( 360 \div 12 = 30^{\circ} \), so interior \( 150^{\circ} \).
3. (Non-calculator.) Each interior angle of a regular polygon is \( 140^{\circ} \). How many sides does it have? [2]
Exterior \( 40^{\circ} \); \( 360 \div 40 = 9 \) sides.
4. (Non-calculator.) Write down the number of lines of symmetry and the order of rotational symmetry of (a) a regular hexagon, (b) a parallelogram (not a rectangle or rhombus). [2]
(a) 6 lines, order 6. (b) 0 lines, order 2.
5. (Non-calculator.) Triangles \(ABC\) and \(PQR\) are similar, with \( AB = 6 \) cm, \( BC = 9 \) cm and \( PQ = 10 \) cm, where \(PQ\) corresponds to \(AB\). Find \(QR\). [2]
\( k = \tfrac{10}{6} \); \( QR = 9 \times \tfrac{10}{6} = 15 \) cm.
6. (Non-calculator.) EXTENDED In triangle \(ABC\), \(D\) is on \(AB\) and \(E\) is on \(AC\), with \(DE\) parallel to \(BC\). \( AD = 4 \) cm, \( AB = 10 \) cm and \( DE = 6 \) cm. Explain why triangles \(ADE\) and \(ABC\) are similar, and find \(BC\). [4]
Angle \(A\) is common; angle \( ADE = \) angle \(ABC\) (corresponding angles, \(DE\) parallel to \(BC\)). Two pairs of equal angles, so the triangles are similar. \( k = \tfrac{10}{4} = 2.5 \), so \( BC = 6 \times 2.5 = 15 \) cm.
7. (Calculator.) EXTENDED Two similar rectangles have lengths in the ratio \( 2 : 5 \). The smaller has area 12 cm². Find the area of the larger. [2]
Area factor \( \left(\tfrac{5}{2}\right)^{2} = 6.25 \); \( 12 \times 6.25 = 75 \) cm².
8. (Non-calculator.) EXTENDED How many planes of symmetry does (a) a cuboid measuring 2 cm by 3 cm by 5 cm have, (b) a prism whose cross-section is an equilateral triangle have? [2]
(a) 3. (b) 4: three planes along the length, each through one edge and the middle of the opposite face, plus one plane cutting the prism in half across its length.

🔗Go deeper — other people’s work

These are external resources, not mine. If one stops working, tell me and everything above it on this page still stands.

  • GeoGebra — drag a transversal across parallel lines and watch the angle pairs stay equal
  • Corbettmaths — similar shapes: area and volume