Year 9
Apply the four congruence conditions (SSS, SAS, ASA, RHS) to identify when triangles must be congruent, and justify congruence in formal proof.
Geometry — congruence and proof
Two triangles are congruent if they are identical in shape and size. There are four conditions that guarantee congruence: SSS (all three sides equal), SAS (two sides and the included angle equal), ASA (two angles and the included side equal), and RHS (right angle, hypotenuse and one other side equal in right-angled triangles). When two triangles are congruent, all corresponding sides and angles are equal.
SSS — all three sides equal
Triangles ABC and DEF have AB = DE = 5 cm, BC = EF = 6 cm, AC = DF = 7 cm. Are they congruent?
If all three sides of one triangle match the three sides of another, the triangles must be congruent. Order of sides doesn't matter as long as they match in length.
SAS — two sides and the included angle
Triangles PQR and XYZ have PQ = XY = 8 cm, angle Q = angle Y = 45°, and QR = YZ = 6 cm. Are they congruent?
The angle must be between the two sides. If the angle is not between them, this is not SAS and the triangles are not necessarily congruent.
ASA — two angles and the included side
Triangles MNO and UVW have angle M = angle U = 50°, MN = UV = 7 cm, and angle N = angle V = 60°. Are they congruent?
The side must be between the two angles. Also, the third angles will be equal automatically (angles sum to 180°).
In a rectangle ABCD, the diagonals AC and BD intersect at point O. Prove that triangles AOB and COD are congruent.
For each pair of triangles, identify which congruence condition applies (or state why they are not necessarily congruent).
Triangles have sides 5, 7, 9 and sides 5, 7, 9 respectively. By which condition are they congruent?
💡 All three sides are equal, so it's Side-Side-Side (SSS).
Two triangles have two sides of 6 cm and 8 cm, with the angle between them being 50° in both. By which condition are they congruent?
💡 Two sides and the included angle (the angle between them) — that's SAS.
Two right-angled triangles have hypotenuse 13 cm and one leg 5 cm. By which condition are they congruent?
💡 Right angle, hypotenuse, and one side — RHS is the condition for right-angled triangles.
Two triangles both have angles 40°, 60° and 80°, but the sides are different. Are they congruent?
💡 Angles alone (AAA) do not guarantee congruence — triangles with the same angles are similar, not necessarily congruent.
Which of these is NOT a valid congruence condition for triangles?
💡 SSA (two sides and a non-included angle) is not sufficient — it can give two different triangles (ambiguous case).
Watch for these when working through the lesson.
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