Question

emily and her classmates are discussing the triangles below. emily says they are similar based on aa similarity. sarah says they are similar based on ssa similarity. elinor says they are similar based on sas similarly. who is correct?

Explanation:

Step1: Check AA similarity

Both triangles are right - angled (one right angle each). For the first triangle with sides \(6\) and \(8\), and the second with sides \(3\) and \(4\). The ratios of the corresponding sides are \(\frac{8}{4}=2\) and \(\frac{6}{3}=2\). Also, the right angles are equal. Since two angles (the right angle and the angle determined by the proportional sides) are equal, AA similarity can be used.

Step2: Check SSA similarity

SSA (Side - Side - Angle) is not a valid similarity criterion for triangles. There are cases where SSA does not guarantee similarity (ambiguous case).

Step3: Check SAS similarity

For SAS similarity, we need two sides in proportion and the included angle equal. Here, the sides are in proportion (\(\frac{8}{4}=\frac{6}{3} = 2\)), but the included angles for the given sides are the right angles. However, the order of sides in SAS for similarity is not just about two sides and an angle, but also the correspondence. But AA is a more straightforward and valid criterion here compared to the incorrect SSA and the less - directly - applicable (in terms of the problem's context of students' statements) SAS (since AA is more basic for right - angled triangles with proportional sides).

Answer:

Emily is correct.