Question

determine if triangle lmn and triangle opq are or are not similar, and, if they are, state how you know. (note that figures are not necessarily drawn to scale.)
answer
the triangles $square$ are similar.
$square$ are not

Explanation:

Step1: List side lengths of triangles

Triangle LMN: $LN=20$, $NM=16$, $ML=21$
Triangle OPQ: $PQ=32$, $QO=40$, $OP=42$

Step2: Order sides from shortest to longest

LMN: $16, 20, 21$
OPQ: $32, 40, 42$

Step3: Calculate side ratios

$\frac{16}{32}=\frac{1}{2}$, $\frac{20}{40}=\frac{1}{2}$, $\frac{21}{42}=\frac{1}{2}$

Step4: Check similarity condition

All corresponding side ratios are equal, so by SSS (Side-Side-Side) similarity criterion, the triangles are similar.

Answer:

The triangles are similar, by the SSS similarity criterion (all corresponding side lengths are in the same ratio of $\frac{1}{2}$).