MathIsimple

Advanced Similar Triangles

Master similarity criteria (AA, SAS, SSS), solve proportions, write similarity proofs, and apply similarity to solve complex geometric problems!

10th Grade
Geometry
~60 min
🎮 Interactive Activity: Similarity Checker

Determine if triangles are similar!

Triangle 1: Angles 60°, 60°, 60°

Triangle 2: Sides 3, 4, 5

🎮 Interactive Activity: Proportion Solver

Use proportions to find missing sides!

Problem:

If ΔABC ~ ΔDEF and AB = 6, DE = 9, BC = 8, find EF
1. Review of Similarity

What Are Similar Triangles?

Two triangles are similar if they have the same shape but may differ in size. This means:

  • All corresponding angles are equal
  • All corresponding sides are proportional
  • The ratio of corresponding sides is the scale factor

Notation: ΔABC ~ ΔDEF means triangle ABC is similar to triangle DEF

Example 1: Identifying Similarity

Triangle 1: Angles 60°, 60°, 60°
Triangle 2: Angles 60°, 60°, 60°

Analysis: All corresponding angles are equal

Conclusion: Triangles are similar by AA (Angle-Angle)

Note: Both are equilateral triangles, so they're always similar

Example 2: Scale Factor

If ΔABC ~ ΔDEF and AB = 4, DE = 6, find the scale factor.

Scale factor: DE/AB = 6/4 = 1.5

Interpretation: ΔDEF is 1.5 times larger than ΔABC

2. Similarity Criteria
3. Solving Proportions
4. Writing Similarity Proofs
5. Important Theorems
6. Real-World Applications
7. Special Cases
8. Problem-Solving Strategies
Frequently Asked Questions

Practice Time!

Practice Quiz
10
Questions
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Correct
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Score
1
Which similarity criterion requires all three angles to be equal?
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2
If two triangles have sides in ratio 2:3, what is the scale factor?
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3
What does SAS similarity require?
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4
If ΔABC ~ ΔDEF and AB = 6, DE = 9, BC = 8, what is EF?
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5
Two triangles are similar if:
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6
What is the scale factor if a triangle's sides are multiplied by 2?
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7
If two triangles are similar, what is true about their areas?
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8
Which theorem states that a line parallel to one side divides the other sides proportionally?
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9
If ΔABC ~ ΔDEF with scale factor 3, and area of ΔABC is 5, what is area of ΔDEF?
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10
What is the ratio of perimeters for two similar triangles with scale factor k?
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