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Understanding Similarity and Congruence in Triangles: Key Concepts and Distinctions

This comprehensive guide explains the fundamental differences between similar and congruent triangles. It outlines the criteria for determining whether two triangles are similar, focusing on equal corresponding angles and proportional side lengths. We also explore real-world applications of these concepts, such as indirect measurement using similar triangles, and how to justify your answer when analyzing triangle relationships. Perfect for students and educators seeking clarity on geometric principles.

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Understanding Similarity and Congruence in Triangles: Key Concepts and Distinctions

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  1. Similar vs. Congruent

  2. 5-3 • Yes • No • No • Yes • No • Yes • No • Yes • Yes • No • No • No • Not proportional • Proportional • Proportional • Not proportional • Proportional • Not proportional • Not proportional • Proportional • Proportional • Not proportional • Proportional • Proportional • No • yes

  3. 5-4 • 8 • 14 • 15 • 7.5 • 28 • 6 • 35 • 20 • 9 • 6 • 2 • 18 • $12,000 • 1c • 67.5 min • 364 mi • 60 days • 18 eggs

  4. Similar or Congruent?

  5. Similar or Congruent?

  6. Similar or Congruent?

  7. Congruent or Similar?

  8. How do we know if two triangles are similar or proportional? BACK NEXT EXIT

  9. Triangles are similar (~) if corresponding angles are equal and the ratios of the lengths of corresponding sides are equal. BACK NEXT EXIT

  10. B C A Interior Angles of Triangles The sum of the measure of the angles of a triangle is 1800. Ð A + Ð B + ÐC =1800

  11. Determine whether the pair of triangles is similar. Justify your answer. Example 6-1b Answer: Since thecorresponding angles have equal measures, the triangles are similar.

  12. This tells us that  ABC and  XYZ are similar and proportional.

  13. Q: Can these triangles be similar?

  14. Answer—Yes, right triangles can also be similar but use the criteria.

  15. Do we have equality? This tells us our triangles are not similar. You can’t have two different scaling factors!

  16. If we are given that two triangles are similar or proportional what can we determine about the triangles?

  17. The two triangles below are known to be similar, determine the missing value X.

  18. A 5 P 10 R 4 d c B Q 6 C In the figure, the two triangles are similar. What are c and d ?

  19. A 5 P 10 R 4 d c B Q 6 C In the figure, the two triangles are similar. What are c and d ?

  20. Sometimes we need to measure a distance indirectly. A common method of indirect measurement is the use of similar triangles. h 6 17 102

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