Question

which rule explains why these scalene triangles are similar?
sss
sas
aa
none of the above; the triangles cannot be proven similar

Explanation:

Step1: Calculate the ratio of corresponding sides

We have two sides of the first triangle \(JH = 45\) and \(HI=30\), and two sides of the second triangle \(SQ = 72\) and \(QR = 48\).
The ratio of \(JH\) to \(SQ\) is \(\frac{45}{72}=\frac{5}{8}\), and the ratio of \(HI\) to \(QR\) is \(\frac{30}{48}=\frac{5}{8}\).

Step2: Check the included angle

The included angle \(\angle H\) in \(\triangle JHI\) and \(\angle Q\) in \(\triangle SQR\) are equal (as indicated by the red arcs).

Since we have two pairs of sides in proportion (\(\frac{JH}{SQ}=\frac{HI}{QR}\)) and the included angles equal (\(\angle H=\angle Q\)), by the SAS (Side - Angle - Side) similarity criterion, the two triangles are similar.

Answer:

SAS