The image you've uploaded is an educational diagram explaining the
rules for determining similar triangles, specifically the
Angle-Angle (AA),
Side-Angle-Side (SAS), and
Side-Side-Side (SSS) similarity criteria. Let's go through each rule and explain how they work.
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🔷 1. Angle-Angle (AA) Similarity Rule
Statement:
> If ∠CAB ≅ ∠GEF and ∠ABC ≅ ∠EFG, then
> △ABC ~ △EFG
Explanation:
- Two triangles are similar if
two pairs of corresponding angles are congruent.
- Since the sum of angles in any triangle is always 180°, if two angles are equal, the third angle must also be equal.
- So, AA guarantees that all three angles are equal → the triangles have the same shape (but possibly different sizes).
-
Note: The order of vertices matters — the correspondence must match the angles.
✔ Conclusion: AA is sufficient to prove similarity.
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🔷 2. Side-Angle-Side (SAS) Similarity Rule
Statement:
> If ∠CAB ≅ ∠FEG and
> AB/EG = AC/EF, then
> △ABC ~ △EFG
Explanation:
- Two sides of one triangle are proportional to two sides of another triangle,
and the included angles between those sides are congruent.
- The "included angle" means the angle formed by the two sides being compared.
- This ensures that the triangles have the same shape because both proportions and angle are preserved.
⚠️ Important: The angle must be
between the two sides (i.e., the included angle).
✔ Conclusion: SAS similarity applies only when the proportionality of two sides and the equality of the included angle are satisfied.
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🔷 3. Side-Side-Side (SSS) Similarity Rule
Statement:
> If AB/EF = BC/FG = AC/EG, then
> △ABC ~ △EFG
Explanation:
- All three pairs of corresponding sides are in proportion.
- This means the triangles are scaled versions of each other.
- Even without knowing the angles, if all side ratios are equal, the triangles must be similar.
✔ Conclusion: SSS similarity confirms similarity based solely on proportional side lengths.
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📌 Summary Table:
| Rule | Condition | Conclusion |
|------|---------|------------|
|
AA | Two pairs of angles are congruent | Triangles are similar |
|
SAS | Two pairs of sides are proportional
and the included angles are congruent | Triangles are similar |
|
SSS | All three pairs of sides are proportional | Triangles are similar |
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✔ Key Takeaways:
-
Similar triangles have the
same shape, but not necessarily the same size.
- The notation △ABC ~ △EFG means triangle ABC is similar to triangle EFG with a vertex correspondence: A ↔ E, B ↔ F, C ↔ G.
- These rules are essential for solving problems involving indirect measurement, scale drawings, shadows, and geometry proofs.
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Let me know if you'd like help applying these rules to a specific problem!
Parent Tip: Review the logic above to help your child master the concept of similar triangles.
