Congruence and similarity compare shapes by size and shape. They are central for triangle proofs, scale drawings, and indirect measurement.

Congruence

Congruent figures have exactly the same shape and size. Corresponding sides and angles are equal.

For triangles, common criteria:

• SSS (three sides equal)
• SAS (two sides and included angle equal)
• ASA / AAS (two angles and one corresponding side equal)
• RHS (right triangle: hypotenuse and one side equal)

Similarity

Similar figures have the same shape but possibly different sizes.

• Corresponding angles are equal.
• Corresponding sides are in one constant ratio (scale factor).

Triangle similarity criteria:

• AA (two angles equal)
• SAS in proportion
• SSS in proportion

Scale factor relationships

If linear scale factor from shape A to B is \(k\):

• Lengths multiply by \(k\).
• Areas multiply by \(k^2\).
• Volumes multiply by \(k^3\).

Using similarity to solve problems

1. Identify corresponding vertices in correct order.
2. Write ratio equation using matching sides.
3. Solve unknown lengths/areas.
4. Check ratio consistency across all pairs.

Congruence and similarity in proofs

Common proof flow:

• Show two triangles are congruent/similar by criteria.
• Conclude equal angles/sides or proportional segments.
• Use result to prove target statement.

Exam strategy

• State criterion explicitly (e.g., “\(\triangle ABC\cong\triangle DEF\) by SAS”).
• Match vertices in correct correspondence order.
• Do not mix congruence and similarity conditions.
• For ratio problems, keep fractions in consistent side order.