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Geometry & measures

Congruence and Similarity

Year 10 · Year 11

  • ✓By the end of this lesson students will be able to identify congruent triangles using SSS, SAS, ASA, and RHS criteria.
  • ✓By the end of this lesson students will be able to understand and apply the properties of similar shapes to find missing lengths.
  • ✓By the end of this lesson students will be able to calculate and use area and volume scale factors for similar 2D and 3D shapes.

Key concepts

Congruent Shapes

Shapes are congruent if they are exactly the same size and shape. One can be transformed into the other by a combination of translations, rotations, and reflections. All corresponding lengths and angles are equal.

Congruent Triangles Criteria

Two triangles are congruent if one of the following conditions is met: 1. SSS (Side-Side-Side): All three corresponding sides are equal. 2. SAS (Side-Angle-Side): Two corresponding sides and the included angle are equal. 3. ASA (Angle-Side-Angle): Two corresponding angles and the included side are equal. 4. RHS (Right-angle-Hypotenuse-Side): A right angle, the hypotenuse, and one other side are equal.

Similar Shapes

Shapes are similar if one is an enlargement or reduction of the other. This means they have the same shape but different sizes. For similar shapes, all corresponding angles are equal, and all corresponding sides are in the same ratio (i.e., they have a constant linear scale factor).

Linear Scale Factor (k)

The linear scale factor (k) is the ratio of corresponding lengths in two similar shapes. It is calculated by dividing a length on the 'new' or 'enlarged' shape by the corresponding length on the 'original' or 'smaller' shape.

k = (new length) / (original length)
Area Scale Factor (k²)

If two shapes are similar with a linear scale factor k, then the ratio of their areas is k². To find the area of the larger shape, multiply the area of the smaller shape by k². To find the area of the smaller shape, divide the area of the larger shape by k².

Area_new / Area_original = k²
Volume Scale Factor (k³)

If two 3D shapes are similar with a linear scale factor k, then the ratio of their volumes is k³. To find the volume of the larger shape, multiply the volume of the smaller shape by k³. To find the volume of the smaller shape, divide the volume of the larger shape by k³.

Volume_new / Volume_original = k³

Key facts to remember

  • 1Congruent shapes are identical in shape and size.
  • 2Similar shapes are identical in shape but different in size (one is an enlargement of the other).
  • 3The four criteria for proving triangle congruence are SSS, SAS, ASA, and RHS.
  • 4For similar shapes, corresponding angles are equal, and corresponding sides are in proportion.
  • 5The linear scale factor (k) is the ratio of corresponding lengths.
  • 6If the linear scale factor is k, the area scale factor is k².
  • 7If the linear scale factor is k, the volume scale factor is k³.
  • 8To find a scale factor, always divide a dimension from the 'new' shape by the corresponding dimension from the 'original' shape.

Worked examples

Example 1

Prove that triangle ABC is congruent to triangle DEF, given that AB = DE, BC = EF, and AC = DF.

IIdentify the given information:
IIAB = DE (Side)
IIIBC = EF (Side)
IVAC = DF (Side)
VSince all three corresponding sides are equal, the triangles are congruent by the SSS criterion.

Answer

Triangle ABC is congruent to triangle DEF (SSS).

Always state the specific congruence criterion used.

Example 2

Triangle PQR is similar to triangle XYZ. PQ = 6 cm, QR = 8 cm, PR = 10 cm. If XY = 9 cm, find the lengths of YZ and XZ.

IIdentify corresponding sides: PQ corresponds to XY, QR corresponds to YZ, PR corresponds to XZ.
IICalculate the linear scale factor (k) using the known corresponding sides:
IIIk = XY / PQ = 9 cm / 6 cm = 1.5
IVFind YZ:
VYZ = QR × k = 8 cm × 1.5 = 12 cm
VIFind XZ:
VIIXZ = PR × k = 10 cm × 1.5 = 15 cm

Answer

YZ = 12 cm, XZ = 15 cm.

Ensure you consistently use the 'new' length divided by 'original' length to find the scale factor.

Example 3

Two mathematically similar cylinders have heights of 4 cm and 10 cm. The volume of the smaller cylinder is 80 cm³. Calculate the volume of the larger cylinder.

ICalculate the linear scale factor (k) from the heights:
IIk = (height of larger cylinder) / (height of smaller cylinder) = 10 cm / 4 cm = 2.5
IIICalculate the volume scale factor (k³):
IVk³ = (2.5)³ = 2.5 × 2.5 × 2.5 = 15.625
VCalculate the volume of the larger cylinder:
VIVolume_larger = Volume_smaller × k³ = 80 cm³ × 15.625 = 1250 cm³

Answer

The volume of the larger cylinder is 1250 cm³.

Remember to cube the linear scale factor for volume calculations.

Common mistakes

  • ✗Confusing congruence with similarity, or vice-versa.
  • ✗Incorrectly identifying corresponding sides or angles in similar shapes.
  • ✗Using the linear scale factor for area or volume calculations without squaring or cubing it.
  • ✗Not stating the correct congruence criterion when proving triangles are congruent.
  • ✗Assuming shapes are similar or congruent without sufficient information or proof.

Exam tips

  • ★Always state the congruence criterion (SSS, SAS, ASA, RHS) when proving triangles are congruent.
  • ★Draw diagrams and label corresponding sides and angles clearly for similar shapes to avoid errors.
  • ★Be careful to use the correct scale factor (linear, area, or volume) for the quantity you are calculating.
  • ★Clearly show your working, especially when calculating scale factors and applying them to find unknown values.

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