A complete guide to congruent and similar triangles — the SSS, SAS, ASA and RHS congruence tests, the AA, SAS and SSS similarity tests, scale factors, and area and volume ratios.
Two shapes can be related in two distinct ways: they can be congruent (identical in every measurement — the same shape and the same size) or similar (the same shape but a different size, scaled up or down uniformly). Both ideas rest on comparing angles and sides carefully, and both are proved using a small set of recognised tests rather than by "eyeballing" a diagram. This guide works through every congruence and similarity test in depth, how each is used to prove geometric properties, and how similarity connects to enlargement, scale factors, and the ratio of areas and volumes.
Two shapes are congruent if they are exactly the same shape and exactly the same size — one could be placed exactly on top of the other after a combination of translation, rotation and/or reflection, with no resizing. For triangles, congruence can be proved without checking all six measurements (three sides and three angles); a small set of tests is enough to guarantee it.
Four standard tests are used to prove two triangles congruent:
| Test | What must match | Notes |
|---|---|---|
| SSS (Side-Side-Side) | All three sides are equal | The strongest and simplest test — if all sides match, the triangle is fully fixed |
| SAS (Side-Angle-Side) | Two sides and the angle between them are equal | The angle must be the included angle, not any angle |
| ASA (Angle-Side-Angle) | Two angles and the side between them are equal | Also written AAS if the equal side is not between the angles, since the third angle can always be found |
| RHS (Right angle-Hypotenuse-Side) | A right angle, the hypotenuse, and one other side are equal | Only valid for right-angled triangles |
Triangle ABC has AB = 7 cm, BC = 9 cm and angle ABC = 62°. Triangle DEF has DE = 7 cm, EF = 9 cm and angle DEF = 62°. Since two sides and the included angle match, the triangles are congruent by SAS.
A common trap is "SSA" — two sides and a non-included angle. This is not a valid congruence test on its own, because it can sometimes produce two different triangles. Always check that the angle you have is the one between the two known sides before using SAS.
Congruence is not just about recognising identical triangles — it is the standard tool for proving geometric properties. Many quadrilateral properties are proved by splitting the shape into two triangles with a diagonal, then showing those triangles are congruent.
In parallelogram ABCD, diagonals AC and BD intersect at point O. Consider triangles AOB and COD. Since AB is parallel to DC, angle OAB = angle OCD and angle OBA = angle ODC (alternate angles). Also AB = DC (opposite sides of a parallelogram). So triangles AOB and COD are congruent by ASA. Therefore AO = OC and BO = OD — the diagonals bisect each other.
Two shapes are similar if they have exactly the same shape but not necessarily the same size — every angle in one shape matches the corresponding angle in the other, and every pair of corresponding sides is in the same ratio (the scale factor). Congruent shapes are a special case of similar shapes where the scale factor is exactly 1.
| Test | What must match |
|---|---|
| AA (Angle-Angle) | Two pairs of angles are equal (the third pair is then automatically equal too) |
| SAS (Side-Angle-Side, similarity form) | Two pairs of sides are in the same ratio, and the included angle is equal |
| SSS (Side-Side-Side, similarity form) | All three pairs of sides are in the same ratio |
Triangle PQR has angle P = 55° and angle Q = 70°. Triangle XYZ has angle X = 55° and angle Y = 70°. Since two pairs of angles match, the triangles are similar by AA (their third angles must also match, since angles in a triangle sum to 180°).
The scale factor of an enlargement between two similar shapes is the ratio of any pair of corresponding lengths: scale factor = (new length) ÷ (original length). A scale factor greater than 1 makes a shape larger; a scale factor between 0 and 1 makes it smaller.
Triangles ABC and DEF are similar, with AB = 6 cm corresponding to DE = 15 cm. If BC = 8 cm, find EF.
Scale factor = 15 ÷ 6 = 2.5. So EF = 8 × 2.5 = 20 cm.
If two similar shapes have a linear scale factor of k, their areas are in the ratio k², and (for similar solids) their volumes are in the ratio k³. This is because area involves two length dimensions multiplied together, and volume involves three.
Two similar cylinders have heights 4 cm and 10 cm. The linear scale factor is k = 10 ÷ 4 = 2.5.
The ratio of their curved surface areas is k² = 2.5² = 6.25. The ratio of their volumes is k³ = 2.5³ = 15.625.
If the smaller cylinder has a volume of 50 cm³, the larger cylinder's volume is 50 × 15.625 = 781.25 cm³.
A frequent mistake is applying the linear scale factor directly to area or volume. Always square the scale factor for area, and cube it for volume — never use k on its own for either.
Similar triangles often appear "hidden" inside a larger diagram — for example, when a line is drawn parallel to one side of a triangle, or when two triangles share a vertex with parallel sides. Recognising the similar triangles lets you set up a ratio equation to find an unknown length.
In triangle ABC, a line DE is drawn parallel to BC, with D on AB and E on AC. AD = 4 cm, DB = 6 cm, and AE = 5 cm. Find EC.
Since DE is parallel to BC, triangle ADE is similar to triangle ABC (AA, since corresponding angles are equal). So AD/AB = AE/AC.
AB = AD + DB = 10 cm. So 4/10 = 5/AC, giving AC = 12.5 cm, so EC = AC − AE = 12.5 − 5 = 7.5 cm.
More advanced problems often use both ideas together: similarity to establish a relationship between two different-sized shapes, and congruence to prove a more specific equality within one of them. A reliable strategy is to mark every angle and side you know, identify any parallel lines (which create equal angles), and check whether a full congruence test or only a similarity test is supported by the given information.
Common errors include: using "SSA" as if it were a valid congruence test (it is not, except in the special RHS case); forgetting that similarity requires all corresponding sides in the same ratio, not just one pair; applying a linear scale factor directly to area or volume instead of squaring or cubing it; and mismatching corresponding vertices when writing a similarity or congruence statement (the order of the letters should reflect which vertices correspond).
| Idea | Key fact |
|---|---|
| Congruent shapes | Identical in shape and size; SSS, SAS, ASA/AAS, RHS prove it for triangles |
| Similar shapes | Identical in shape, different in size; AA, SAS, SSS (similarity form) prove it for triangles |
| Scale factor k | Ratio of any pair of corresponding lengths |
| Area ratio | k² |
| Volume ratio | k³ |
Every congruence or similarity problem reduces to identifying which test applies, then either concluding the shapes match exactly (congruence) or setting up a ratio to find a missing length, area or volume (similarity).
This article is original EDUSAMBAM educational writing. It is designed as a broad congruence and similarity resource covering the core tests and applications needed for confident geometric proof and problem-solving. Exact examination requirements can vary between examination boards and syllabuses, so students should also compare their work with the specification and past-paper requirements of their own board.
Recommended study approach: learn the four congruence tests and three similarity tests by name and condition, practise identifying which one a diagram supports, then work through area and volume scale factor problems until squaring and cubing k becomes automatic.
20 questions covering congruence tests, similarity tests, scale factors, and area and volume ratios for similar shapes. Answer every question, then submit to see your score instantly.