Question

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

Explanation:

Step1: Check SSS similarity

SSS similarity requires \(\frac{a}{a'}=\frac{b}{b'}=\frac{c}{c'}\).
For \(\triangle EFG\) with sides \(40,50,65\) and \(\triangle HIJ\) with sides \(44,41,68\).
\(\frac{40}{44}=\frac{10}{11}\approx0.91\), \(\frac{50}{41}\approx1.22\), \(\frac{65}{68}\approx0.96\).
Since \(\frac{40}{44}
eq\frac{50}{41}
eq\frac{65}{68}\), SSS similarity does not hold.

Step2: Check SAS similarity

SAS similarity requires \(\frac{a}{a'}=\frac{b}{b'}\) and the included angles equal.
There is no information about included angles being equal, and \(\frac{40}{44}
eq\frac{65}{68}\) (from above). So SAS similarity does not hold.

Step3: Check AA similarity

AA similarity requires two pairs of equal angles.
There is no information about angles being equal.

Answer:

none of the above; the triangles cannot be proven similar