Congruence and Similarity

HSG.SRT.B.5

High School Geometry

Learning Objectives

• Select appropriate criteria (ASA, SAS, AA, etc.) for proofs.
• Use congruence to prove properties of figures (like parallelograms).
• Use similarity to solve indirect measurement problems.
• Combine congruence and similarity in multi-step problems.
• Justify every step in a geometric proof.

The Geometric Toolbelt

Congruence (Same Size & Shape)

• SSS, SAS, ASA, AAS, HL
• Use when: Proving segments or angles are equal.

Similarity (Same Shape, Scale Factor )

• AA, SAS~, SSS~
• Use when: Finding lengths, ratios, or indirect measures.

The Decision Tree

1. What is the goal?

   • Proving Equality? Congruence
   • Finding a Length? Similarity

2. What do I know?

   • Parallel lines? Alt. Interior Angles.
   • Shared parts? Reflexive Property.
   • Right angles? HL or Pythagorean.

Proving Properties: Parallelograms

Prove: Opposite sides of a parallelogram are congruent.

Strategy:

1. Draw diagonal .
2. Identify alternate interior angles (from parallels).
3. Identify shared side .
4. ASA Congruence CPCTC.

Indirect Measurement (Similarity)

Problem:

• Tree shadow: 50 ft.
• Person (6 ft) shadow: 4 ft.
• Find Tree Height (h).

Strategy:

• AA Similarity (Right angles + Sun angle).
• Proportion: .

Calculation

Multi-Step: The Midsegment

Given: are midpoints of .
Prove: and .

Step 1: (SAS~).

• shared.
• .

Multi-Step: The Conclusion

Step 2: Corresponding angles are equal.

• .

Step 3: Proportional sides.

• .

Proving Quadrilaterals

Given: and .
Prove: is a parallelogram (so ).

Strategy:

• by SAS.
• by CPCTC.
• Alt. Interior Angles Equal .

Geometric Mean (Right Triangles)

Altitude to hypotenuse creates 3 Similar Triangles.

Theorem:
The altitude is the geometric mean of the hypotenuse segments.

Application: Finding height from base segments alone.

Writing Proofs: Tips

1. Draw and Mark: Never prove in your head. Mark the diagram!
2. State the Reason: Every step needs a "Why." (Given, Definition, Theorem).
3. CPCTC: Only use this after you prove Congruence.
4. Similarity Ratio: Only use this after you prove Similarity.

Common Pitfalls

• Assuming Congruence by Looks: "It looks isosceles!" (Forbidden).
• Using SSA: This is not a valid criterion.
• Mixing Up Ratios: . Be consistent!

Real-World: Engineering

• Congruence: Manufacturing parts that must fit together perfectly.
• Similarity: Creating scale models to test wind resistance or structural loads.

Geometry builds the world.

Summary

1. Choose Wisely: Congruence for equality, Similarity for measurement.
2. Criteria: Know your SAS, ASA, SSS, AA.
3. Justify: Every step must have a reason.
4. Combine: Use algebra + geometry for multi-step proofs.

Next Steps

Trigonometry

Defining Sine, Cosine, and Tangent using similar right triangles.

Coordinate Proofs

Using algebra on a grid to prove these same theorems.