Congruence

Congruent shapes are identical: the same shape and the same size, so all their matching sides and angles are equal. Two triangles are congruent when they match by one of four conditions: SSS, SAS, ASA (or AAS) or RHS.

Key facts

  • Congruent shapes have equal matching sides and equal matching angles.
  • Triangles are congruent if they match by SSS, SAS, ASA (or AAS) or RHS. AAA is not enough.
  • To prove two triangles congruent, give three matching facts and name the condition they meet.

Worked examples

Example 1

Two triangles have two pairs of equal sides and the angles between those sides are equal. Are they congruent?

  1. Two sides and the included angle fix the triangle.
  2. So the triangles are congruent by SAS.

Answer: Yes (SAS)

Example 2

Two right-angled triangles both have a hypotenuse of 13 cm and another side of 5 cm. Are they congruent? Give the reason.

  1. Both have a right angle, the same hypotenuse and one other side the same.
  2. That is the RHS condition, so they are congruent.

Answer: Yes (RHS)

Example 3

Two triangles both have angles of 50° and 70°, and the side between those two angles is 8 cm in both. Are they congruent?

  1. Two angles and the side between them match.
  2. That is the ASA condition, so they are congruent.

Answer: Yes (ASA)

Example 4

Two triangles both have angles of 40°, 60° and 80°. The shortest side of one is 3 cm; the shortest side of the other is 6 cm. Are they congruent?

  1. Equal angles (AAA) only fix the shape, not the size.
  2. The sides are different, so the triangles are not congruent.

Answer: No: they are similar, not congruent

Example 5

Triangles PQR and XYZ are congruent, with P matching X, Q matching Y and R matching Z. PQ = 7 cm and angle Q = 52°. Find XY and angle Y.

  1. Matching sides of congruent triangles are equal. XY = PQ = 7 cm
  2. Matching angles are equal too. ∠Y = ∠Q = 52°

Answer: XY = 7 cm, angle Y = 52°

Example 6

ABCD is a parallelogram. Prove that triangles ABC and CDA are congruent.

  1. Opposite sides of a parallelogram are equal. AB = CD, BC = DA
  2. AC is a side of both triangles. AC = CA
  3. Three pairs of equal sides: the triangles are congruent (SSS).

Answer: SSS: AB = CD, BC = DA and AC is shared