Lesson on Similar Triangles

Key Topics

• Similarity Theorems
• Algebraic Proofs
• Geometric Proofs

Similarity Theorems

1. AA (Angle-Angle) Similarity

• Two triangles are similar if they have two pairs of congruent angles.
• The third angle theorem confirms the third angle is congruent if two are congruent.
• Example: Equilateral triangles (e.g., small and large 60°, 60°, 60° triangles) are similar.

2. SSS (Side-Side-Side) Similarity

• Triangles are similar if all three pairs of corresponding sides are proportional.
• Example: A 3-4-5 triangle and a 6-8-10 triangle are similar with a 1:2 ratio.

3. SAS (Side-Angle-Side) Similarity

• Requires two sets of proportional sides and a congruent included angle.
• Example: If sides have proportionality like 3:3 and 4:4, with a congruent angle, triangles are similar.

Proving Triangle Similarity

Diagrams and Proportions

• Similarity statements must match each letter with corresponding vertices (e.g., A with D, B with E, etc.).
• Use proportions to solve for unknowns, e.g., relating sides of different triangles (e.g., small triangle's side over larger triangle’s side).
• Example Problem: Solving x using proportions of triangle sides.

Checking Proportionality

• Smallest side of one triangle should relate to the smallest side of another to check proportionality.
• Example: 6 corresponds with 4, 7.5 with 5, and 9 with 6.

Using Cross Products

• Solve for variables using proportionality and cross multiplication.
• Example: Solving 5/15 = 7/21 to verify similarity.

Geometric Proofs

Proof Strategies

• Given Information: Use given right angles and proportional sides.
• Proving Similarity: Use AA, SSS, SAS theorems after confirming congruent angles or proportional sides.

Example Proofs

• Proof 1: Show XYZ and ABC are similar using right angles and one pair of proportional sides.
• Proof 2: Use trapezoid properties to prove proportional sides in triangles by showing parallel lines (trapezoid sides) create transversals.

Important Concepts

• Corresponding Sides: Congruent triangles imply proportional corresponding sides.
• Transversals in Trapezoids: Create congruent angles aiding in establishing similarity.

Conclusion

• A thorough understanding of similarity theorems and methods to prove triangle similarity is crucial in geometry.
• Application of algebraic and geometric proofs solidifies comprehension of similar triangles.