Similarity & Congruence: Scale Factors& Proof.

What makes two shapes similar?

Two shapes are similar if they have exactly the same shape but may differ in size. For similar shapes, all corresponding angles are equal and all corresponding sides are in the same ratio. That constant ratio is called the linear scale factor (often denoted k).

For example, if triangle A has sides 3 cm, 4 cm, 5 cm and triangle B has sides 6 cm, 8 cm, 10 cm, then every side of B is exactly twice the corresponding side of A. The linear scale factor from A to B is 2, and the triangles are similar.

To find a missing side in similar shapes, set up a ratio between two pairs of corresponding sides, then cross-multiply to solve. Always make sure you match corresponding sides correctly by using the angles to identify which sides correspond.

What are the four congruence conditions?

Two triangles are congruent if they satisfy any one of these four conditions:

Note that SSA (two sides and a non-included angle) is not a valid congruence condition because it can produce two different triangles (the ambiguous case). Similarly, AAA only proves similarity, not congruence.

How do linear, area, and volume scale factors relate?

This is the single most important relationship in the similarity topic. If two shapes are similar with linear scale factor k, then:

To go from an area scale factor to a linear scale factor, take the square root. To go from a volume scale factor to a linear scale factor, take the cube root. These reverse operations are tested just as often as the forward ones.

How do you prove similarity and congruence?

At IGCSE Higher tier, you may be asked to write a formal geometric proof that two triangles are similar or congruent. The key steps are:

Proving similarity

1. Step 1: Identify the two triangles you are comparing.
2. Step 2: Show that two pairs of corresponding angles are equal (the third pair follows automatically since angles in a triangle sum to 180 degrees). Alternatively, show all three pairs of corresponding sides are in the same ratio.
3. Step 3: State your conclusion: "Therefore triangle X is similar to triangle Y (AA similarity)" or "(SSS similarity)".

Proving congruence

1. Step 1: Identify the two triangles.
2. Step 2: List three pieces of matching information that satisfy one of the four conditions (SSS, SAS, ASA, or RHS).
3. Step 3: State the condition used and your conclusion.

Always give geometric reasons for each statement. For example, say "angle ABC = angle DEF (alternate angles, AB parallel to DE)" rather than simply listing the equal angles.

Worked Examples

Common IGCSE Exam Mistakes

Exam Tips for Similarity & Congruence