Congruence & Similarity | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Congruence

What does congruent mean?

• In maths, two shapes are congruent if they are identical in shape and size. One may be a reflection, rotation, or translation of the other.
• One shape being an enlargement of the other means they are not identical in size, hence not congruent.

How do we prove that two shapes are congruent?

• To prove that two shapes are congruent, you must demonstrate that they have the same shape and size.
• It is not necessary to show that they are oriented in the same direction.
• When a shape is reflected, rotated, or translated, its image remains congruent to the original shape.

Similarity

• Two shapes are similar if they have the same shape and their corresponding sides are proportional.
  • One shape is a scaled version of the other.
• If two triangles of different sizes have identical angles, they are similar.
  • Other shapes can have the same angles but may not be similar.

How do we prove that two shapes are similar?

• To demonstrate that two non-triangular shapes are similar, you need to show that their corresponding sides are proportional.
  • Find the scale factor by dividing the length of one side by the length of the corresponding side on the other shape.
• If the scale factor is consistent for all corresponding sides, the shapes are similar.
• If one shape can be proven to be an enlargement of the other, then the two shapes are similar.

How do we prove that two triangles are similar?

• To prove that two triangles are similar, you only need to show that their angles are the same.
• This can be achieved by using angle properties, such as identifying isosceles triangles, vertically opposite angles, and angles on parallel lines.
• The triangles may appear different and may be oriented differently, so focus on identifying the angles.
  • It can be helpful to sketch both triangles next to each other and facing the same direction.
• If asked to prove that two triangles are similar, state that corresponding angles in similar triangles are equal and provide a reason for each corresponding equal angle.
  • Often, the triangles may be positioned opposite each other in an hourglass shape, so look for the vertically opposite, equal angles.