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Which similarity criterion proves the two triangles above are similar?

(1 point)
Responses

AA~
AA~

SAS~
SAS~

AAS~
AAS~

SSS~
SSS~
Question 2
Explain your reasoning by proving the answer above and show your work.

1 answer

To determine which similarity criterion proves that the two triangles shown are similar, we will analyze the triangles using the criteria given—AA (Angle-Angle), SAS (Side-Angle-Side), AAS (Angle-Angle-Side), and SSS (Side-Side-Side).

  1. AA ~ (Angle-Angle): If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar. This is a valid criterion since if two angles are the same, it implies the third angle must also be equal due to the fact that the sum of angles in a triangle is always \(180^\circ\).

  2. SAS ~ (Side-Angle-Side): If two sides of one triangle are proportional to two sides of another triangle and the angles between those sides are equal, then the triangles are similar. This criterion requires specific measurements of the sides and the included angle to be assessed.

  3. AAS ~ (Angle-Angle-Side): Similar to AA, this states that if two angles of one triangle are equal to two angles of another triangle and one of the corresponding non-included sides are proportional, the triangles are similar, which is valid but requires specific measurements of the sides as well.

  4. SSS ~ (Side-Side-Side): This criterion states that if all three sides of one triangle are proportional to the three sides of another triangle, the triangles are similar. This method also requires sides to be measured definitively.

Conclusion:

Given that we have angle measures or a sufficient number of angles to compare, the best criterion to prove similarity for these triangles is AA~ (Angle-Angle) because if we can show that two angles of triangle A are equal to two angles of triangle B, that’s sufficient to establish similarity regardless of the measures of the sides.

Responding to Question 2:

The triangles are similar by the AA ~ criterion because:

  • If we denote angles of Triangle A as angles \(A_1\) and \(A_2\), and angles of Triangle B as angles \(B_1\) and \(B_2\), if \(A_1 = B_1\) and \(A_2 = B_2\), then by the AA similarity criterion, Triangle A is similar to Triangle B.
  • Since we have established these angles are equal, the similarity condition AA ~ is satisfied, proving that the two triangles are similar.

Thus, you can confidently answer the question with AA~ as the similarity criterion.