Congruence and Similarity

Congruence and similarity are key concepts in Edexcel GCSE Mathematics. Two shapes are congruent if they are exactly the same shape and size. Two shapes are similar if they are the same shape but possibly different sizes — all corresponding angles are equal, and corresponding sides are in the same ratio. Higher tier students must also apply area and volume scale factors for similar shapes.

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Key Vocabulary

Term	Meaning
Congruent	Identical in shape and size (one can be mapped to the other by reflection, rotation or translation)
Similar	Same shape, different size (one can be mapped to the other by an enlargement)
Scale factor	The multiplier that converts one length to the corresponding length in a similar shape
Corresponding	In matching positions in two shapes

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Congruent Triangles

To prove two triangles are congruent, you must show one of these four conditions:

Condition	What you show	Abbreviation
Side-Side-Side	All three sides are equal	SSS
Side-Angle-Side	Two sides and the included angle are equal	SAS
Angle-Side-Angle	Two angles and the included side are equal	ASA
Right angle-Hypotenuse-Side	Both are right-angled, with hypotenuse and one other side equal	RHS

Important: "Angle-Angle-Angle" (AAA) does NOT prove congruence — it only proves similarity.

Worked Example 1

In triangles ABC and DEF: AB = DE = 5 cm, angle BAC = angle EDF = 40°, AC = DF = 7 cm.

AB = DE (given), angle BAC = angle EDF (given), AC = DF (given). The equal angle is between the two equal sides, so triangles are congruent by SAS.

Worked Example 2

In triangles PQR and XYZ: angle P = angle X = 50°, PQ = XY = 6 cm, angle Q = angle Y = 70°.

Angle P = angle X (given), PQ = XY (given), angle Q = angle Y (given). The equal side is between the two equal angles, so triangles are congruent by ASA.

Worked Example 3

In triangles ABC and DEF: angle B = angle E = 90°, AC = DF = 13 cm, AB = DE = 5 cm.

Both have a right angle, equal hypotenuse (AC = DF = 13), and one equal side (AB = DE = 5). Triangles are congruent by RHS.

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Proving Congruence in Context

In exam questions, you often need to prove triangles within a larger shape are congruent.

Worked Example 4

ABCD is a parallelogram. Prove that triangles ABD and CDB are congruent.

AB = CD (opposite sides of a parallelogram are equal) AD = CB (opposite sides of a parallelogram are equal) BD = DB (common side) Therefore, triangle ABD is congruent to triangle CDB by SSS.

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Similar Shapes

Two shapes are similar if:

• All corresponding angles are equal, AND
• All corresponding sides are in the same ratio (the scale factor).

Finding the Scale Factor

Scale factor (SF) = length on larger shape / corresponding length on smaller shape

Worked Example 5

Two similar triangles have sides 3 cm, 4 cm, 5 cm and 6 cm, 8 cm, 10 cm.

SF = 6/3 = 8/4 = 10/5 = 2

The second triangle is an enlargement of the first by scale factor 2.

Finding Missing Lengths in Similar Shapes

Worked Example 6

Two similar rectangles have widths 4 cm and 10 cm. The smaller rectangle has length 6 cm. Find the length of the larger rectangle.

SF = 10/4 = 2.5 Length = 6 x 2.5 = 15 cm

Worked Example 7

Triangles ABC and DEF are similar. AB = 8 cm, DE = 12 cm, BC = 6 cm. Find EF.

SF = 12/8 = 1.5 EF = 6 x 1.5 = 9 cm

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Similar Triangles in Context

Look for similar triangles when:

• Two triangles share an angle, and both have a pair of parallel sides.
• Two triangles have two pairs of equal angles (AA similarity).

Worked Example 8

In triangle PQR, a line ST is drawn parallel to QR where S is on PQ and T is on PR. PS = 4 cm, SQ = 6 cm, QR = 15 cm. Find ST.

Triangles PST and PQR are similar (AA — shared angle P and corresponding angles from parallel lines). PQ = PS + SQ = 4 + 6 = 10 cm SF = PS/PQ = 4/10 = 2/5 ST = QR x (2/5) = 15 x 2/5 = 6 cm

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Area and Volume Scale Factors [H]

When two shapes are similar with linear scale factor k:

Quantity	Scale factor
Length	k
Area	k²
Volume	k³

Worked Example 9 [H]

Two similar cylinders have heights 5 cm and 15 cm. The smaller cylinder has surface area 80 cm². Find the surface area of the larger cylinder.

Linear SF = 15/5 = 3 Area SF = 3² = 9 Surface area = 80 x 9 = 720 cm²

Worked Example 10 [H]

Two similar cones have volumes 24 cm³ and 192 cm³. The larger cone has height 16 cm. Find the height of the smaller cone.

Volume SF = 192/24 = 8 Linear SF = cube root of 8 = 2 Height of smaller = 16/2 = 8 cm

Worked Example 11 [H]

Two similar bottles have heights in the ratio 3 : 5. The smaller bottle holds 270 ml. How much does the larger bottle hold?

Linear ratio = 3 : 5, so volume ratio = 27 : 125 Volume of larger = 270 x (125/27) = 1250 ml

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Common Mistakes and Misconceptions