Congruence of ACB & CED: Similarity Chapter

  • Context: MHB 
  • Thread starter Thread starter slwarrior64
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SUMMARY

The discussion confirms that triangles ACB and CED are congruent, specifically focusing on the congruence of angle ACB and angle CED. Triangle ABC is established as a right triangle, while triangle CED's right angle status is implied but not explicitly stated. The congruence of angles CAB and DCE leads to the conclusion that angle ACE is also a right angle. Consequently, triangle ACE is identified as a right triangle with legs measuring 7 and 24, allowing the calculation of length AE using the Pythagorean theorem.

PREREQUISITES
  • Understanding of triangle congruence criteria
  • Knowledge of right triangle properties
  • Familiarity with the Pythagorean theorem
  • Basic geometric angle relationships
NEXT STEPS
  • Study triangle congruence criteria in detail
  • Explore properties of right triangles
  • Practice Pythagorean theorem applications
  • Investigate geometric angle relationships in various triangle configurations
USEFUL FOR

Students of geometry, mathematics educators, and anyone interested in understanding triangle properties and congruence relationships.

slwarrior64
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Only use that ACB and CED are congruent, not any other symbols in the diagram.
 
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I don't think "similarity" is useful here (and the two triangles are definitely NOT "congruent". We are given that triangle ABC is a right triangle. (It is not specifically said that triangle CED is a right triangle but if angle CDE is not a right angle this problem is un-doable.) Angle ACB is congruent to angle CED. Since both triangles are right angles it follows that angles CAB and DCE are congruent and then that angle ACE is a right angle! So triangle ACE is a right triangle with legs of length 7 and 24. The length of AE follows from the Pythagorean theorem.
 

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