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Sec 2 Congruence and Similarity | Triangle Criteria, Scale Factor and Corresponding Sides

Secondary 2 Congruence and Similarity teaches students to decide when two shapes are exactly the same size and shape, and when they have the same shape but different scale. The difference is fundamental: congruent figures preserve all corresponding lengths; similar figures preserve angles and proportional side lengths.

Students searching for congruence and similarity, Sec 2 geometry, similar triangles or scale factor often match sides by visual position rather than logical correspondence. The durable method is match vertices → identify evidence → state criterion → build corresponding ratios → solve.

This page is the canonical Congruence and Similarity owner under the Secondary Mathematics Topic Library.

Quick answer: congruent versus similar

  • Congruent: same shape and same size.
  • Similar: same shape, corresponding angles equal, corresponding lengths in a constant ratio.

Correspondence comes first

If triangle ABC corresponds to triangle PQR, then A↔P, B↔Q and C↔R.

That determines which sides and angles can be compared.

Do not assume the leftmost side corresponds to the leftmost side in another rotated diagram.

Triangle congruence criteria

  • SSS: three corresponding sides equal.
  • SAS: two sides and included angle equal.
  • ASA/AAS: sufficient corresponding angles and a side to fix size.
  • RHS: right angle, hypotenuse and one corresponding side equal for right triangles.

The exact criterion terminology used by schools can vary slightly, but the mathematical evidence remains the same.

Worked congruence example

Triangle ABC has sides 5,7,8. Triangle PQR has corresponding sides 5,7,8.

By SSS, the triangles are congruent.

Therefore corresponding angles are equal too.

Similarity

Similar triangles have equal corresponding angles and proportional corresponding sides.

Example: one triangle has sides 3,4,5 and another 6,8,10. Scale factor from first to second = 2.

Scale factor

Scale factor = new corresponding length ÷ original corresponding length.

Every corresponding length changes by the same factor.

Worked similarity example

Triangles are similar. Small triangle side 6 corresponds to large triangle side 15. Another small side is 8. Find the corresponding large side.

Scale factor = 15/6 = 2.5.

Large side = 8×2.5 = 20.

Area scale factor

If length scale factor = k, then area scale factor = k².

Example: length factor 3 → area factor 9.

This is because area scales in two dimensions.

Volume scale factor

For similar solids, volume scale factor = k³.

Example: length factor 2 → volume factor 8.

This is a later extension of the same dimensional reasoning.

Angles and similarity

Equal corresponding angles establish the same shape. For triangles, angle information can be sufficient to prove similarity because the third angle is then fixed automatically.

Once similarity is established, side ratios become valid.

Worked example using parallel lines

A line parallel to one side of a triangle creates a smaller triangle inside the original. Corresponding angles are equal because of parallel-line angle properties, so the triangles are similar.

This is a common route from Sec 1 angle reasoning into Sec 2 similarity.

Congruence is similarity with scale factor 1

Every pair of congruent shapes is also similar, with scale factor 1. But similar shapes are not necessarily congruent because their sizes may differ.

Common errors

  • matches non-corresponding sides
  • uses one equal angle to claim similarity
  • uses SSA as automatic congruence evidence
  • uses length scale factor directly for area
  • reverses scale factor direction halfway through
  • assumes diagrams are drawn to scale

The correspondence table

For difficult diagrams, write:

  • A ↔ P
  • B ↔ Q
  • C ↔ R

Then list corresponding sides underneath. This prevents ratio inversion.

The similarity audit

  1. Which vertices correspond?
  2. What evidence proves same shape?
  3. What is the scale-factor direction?
  4. Are the side ratios written in the same order?
  5. If area/volume is involved, did the scale factor get squared/cubed?

Practice

Practice 1

Question: Sides 3,4,5 vs 6,8,10

Answer: similar, scale factor 2

Practice 2

Question: Length factor 4

Answer: area factor 16

Practice 3

Question: Length factor 3

Answer: volume factor 27

Practice 4

Question: Two triangles same three side lengths

Answer: congruent by SSS

Practice 5

Question: Right triangles same hypotenuse and one leg

Answer: congruent by RHS

Frequently asked questions

What is the difference between congruent and similar?

Congruent means same shape and size; similar means same shape with proportional size.

What should I do first in a similarity question?

Match corresponding vertices and sides.

How does scale factor affect area?

Area changes by the square of the length scale factor.

Can rotated shapes still be congruent or similar?

Yes. Orientation does not change the geometric relationship.

Where does this sit in Atlas?

This is the canonical Sec 2 Congruence and Similarity owner under Secondary Mathematics Topic Library.

The final Similarity rule

Match correspondence before calculating anything. Once the correct vertices and sides are paired, congruence and similarity become evidence-and-scale problems rather than visual guesses.

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