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Using any of the three triangle similarities of corresponding angles or side lengths, explain why the two triangles are similar. Reference the triangle similarity criteria of choice (SAS or SSS) explain your reasoning in your answer, and show work for any needed computations.

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To determine whether two triangles are similar, we can use one of the triangle similarity criteria: Side-Angle-Side (SAS) similarity or Side-Side-Side (SSS) similarity. Below, I’ll explain using both criteria how we might prove that two triangles, say Triangle ABC and Triangle DEF, are similar.

Step-by-Step Explanation:

  1. Identify Corresponding Angles or Sides: Let's assume we know:

    • Angle A corresponds to Angle D.
    • Angle B corresponds to Angle E.
    • Angle C corresponds to Angle F.

  2. Using the Angle-Angle (AA) Criterion: If we establish that:

    • \( \angle A = \angle D \)
    • \( \angle B = \angle E \)

    Then we can conclude that the two triangles are similar by the Angle-Angle (AA) similarity criterion.

  3. Using the Side-Side-Side (SSS) Criterion: Suppose we find the following ratios of corresponding sides:

    • \( \frac{AB}{DE} = k \)
    • \( \frac{BC}{EF} = k \)
    • \( \frac{CA}{FD} = k \)

    If all corresponding side lengths are proportional (i.e., have the same ratio, \( k \)), then by the SSS similarity criterion, Triangle ABC is similar to Triangle DEF.

  4. Using the Side-Angle-Side (SAS) Criterion: In case we have a scenario where:

    • The length of two sides of Triangle ABC are proportional to the lengths of two sides of Triangle DEF (let's denote lengths as \( AB, AC \) and \( DE, DF \)):
      • \( \frac{AB}{DE} = k \)
      • \( \frac{AC}{DF} = k \)
    • The angle between those sides is equal:
      • \( \angle ABC = \angle EDF \)

    If these conditions are satisfied, then by the SAS similarity criterion, Triangle ABC is similar to Triangle DEF.

Conclusion:

By establishing our criteria based on the angles and side lengths, we can use either the AA, SSS, or SAS similarity criterion to conclude that the two triangles are similar. In your specific case, you would need to apply the same reasoning with actual measures of angles and lengths to reach a conclusion about the similarity of the triangles you are analyzing.