Similarity & Congruence

TipLearning Objectives
  • Determine whether triangles are similar or congruent.
  • Use criteria: AA~, SSS~, SAS~, SSS, SAS, ASA.
  • Solve for missing sides in proportional triangles.

Key Ideas

Similar Triangles (∼)

  • Same shape, proportional sides.
  • Angle-Angle (AA) guarantees similarity.

Congruent Triangles (≅)

  • Same shape and size.
  • Proven by SSS, SAS, ASA (and AAS).

Corresponding sides matter:

Set proportions correctly: \[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}. \]

Pair of similar triangles with side labels \(a,b,c\) and scaled sides \(ka,kb,kc\), placed cleanly away from triangle edges.

Triangle Criteria Summary

Criterion Sim? Cong? Notes
SSS ✔️ ✔️ Similarity: sides proportional. Congruence: sides equal.
SAS ✔️ ✔️ Similarity: included angle with proportional adjacent sides.
ASA ✔️ ✔️ Works for both; similarity also works with AAA.
AAS ✔️ ✔️ Same reasoning as ASA; AAA enough for similarity.
AAA ✔️ ❌ Determines shape only, not size → no congruence.
HL ❌ ✔️ Right triangle hypotenuse–leg criterion (congruence only).
SSA ❌ ❌ Ambiguous case; does not guarantee similarity or congruence.

Common Problem Types

Proving Similarity

Use AA or proportional sides.

Solving for Missing Sides

Cross-multiply proportional sides.

Congruence Proofs

Identify correct correspondence.

Scale Factor

Determine enlargement or reduction.

Area Relations

If scale factor is k, area scales by \(k^2\).

Strategies

  • Match vertices carefully.
  • Use angle markings.
  • Check ratios before solving.
  • Look for parallel lines → create similar triangles.

Worked Examples

Example 1 — Similarity

Triangles with two equal angles → similar.

Example 2 — Proportion

If triangles are similar and
\(\frac{AB}{DE}=\frac{4}{10}\),
then scale factor = 0.4.

WarningCommon Mistakes
  • Mismatching order of vertices.
  • Using wrong sides in ratios.
  • Confusing similarity (∼) with congruence (≅).

Practice Problems

  1. Given two angles match, are triangles similar?
  2. If corresponding sides are 6 and 9, find scale factor.
  3. If scale factor = 2, by what factor does area change?
  4. In similar triangles, if small side = 4 and scale factor = 3, find large side.
  1. Yes (AA).
  2. \(6/9 = 2/3\).
  3. Area multiplies by \(2^2 = 4\).
  4. \(4 × 3 = 12\).

Summary

  • Similar → proportional sides; congruent → identical.
  • AA is enough for similarity.
  • Scale factor determines side and area relationships.
  • Always match vertices in order.
  • Scale factor applies linearly; area uses \(k^2\).