0580

Congruence and Similarity

Lengths, Areas and Volumes

Same shape, different size

  • Two shapes are similar when enlarging one of them by some scale factor would produce the other, which means their angles match and every pair of corresponding sides shares the same ratio
  • Papers say "mathematically similar" rather than just "similar", and that phrasing is the signal to look for a scale factor
  • Congruence is the special case of similarity where the scale factor is 1, so congruent shapes are similar but similar shapes are usually not congruent

Proving it

  • For triangles, equal angles are enough: if the three angles match, the sides are automatically in proportion
    • State each pair of equal angles and give the geometrical reason, exactly as in an angles question
    • The reasons are usually vertically opposite angles, alternate or corresponding angles on parallel lines, or angles shared by both triangles
  • For shapes that are not triangles, equal angles are not enough, so show that every pair of corresponding sides gives the same scale factor
  • Similar triangles frequently overlap on the diagram or sit point to point in an hourglass, and redrawing them separately, facing the same way, prevents mismatching the sides
Triangle ABC with a line QP drawn across it parallel to the base CB, so the small triangle AQP sits inside the large triangle ACB and the two share the angle at A, making them similar but overlapping on the diagram
Source: Similar lengths by Save My Exams
Worked example

Showing that two triangles are similar

In triangle ABC, the line DE is parallel to BC, with D on AB and E on AC. Show that triangle ADE is similar to triangle ABC.

Triangle ABC with A at the top, B at the bottom left and C at the bottom right. The point D lies on AB and E lies on AC, and the line DE is drawn across the triangle. DE and BC each carry a single tick mark to show they are parallel. No angles or lengths are given. The figure is marked NOT TO SCALE.

Solution:

  • Similarity needs every pair of corresponding angles to be equal, so take them one pair at a time and name the reason each time
  • Angle A is shared by both triangles, so angle DAE = angle BAC — the angle is common
  • DE is parallel to BC, and AB crosses both, so angle ADE = angle ABC — corresponding angles are equal
  • For the same reason with AC as the crossing line, angle AED = angle ACB
  • All three pairs of corresponding angles are equal
  • Therefore triangle ADE is similar to triangle ABC
  • Naming the reason beside each pair is what earns the marks; listing the equal angles without reasons does not
  • Write the letters in matching order, so that A pairs with A, D with B and E with C