Question

1. sss

triangle cde with sides: ce=96, de=80, cd=64
triangle lmn with sides: lm=54, mn=36, ln=45
handwritten ratios: 96/54 = 64/45 = 80/36

Explanation:

Step1: Identify side lengths

For $\triangle CDE$: $CD = 64$, $CE = 96$, $DE = 80$.
For $\triangle LMN$: $LN = 45$, $LM = 54$, $MN = 36$.

Step2: Check ratios of corresponding sides

Calculate $\frac{CE}{LM}=\frac{96}{54}=\frac{16}{9}$, $\frac{CD}{LN}=\frac{64}{45}$ (Wait, correction: Wait, the user’s handwritten ratio has $\frac{96}{54}$, $\frac{64}{45}$, $\frac{80}{36}$. Let's simplify each:
$\frac{96}{54}=\frac{16}{9}$? No, $96\div6 = 16$, $54\div6 = 9$? Wait, $96\div 6=16$, $54\div6 = 9$? Wait, no, $96\div 12 = 8$, $54\div12 = 4.5$? Wait, the handwritten ratio is $\frac{96}{54}=\frac{64}{45}=\frac{80}{36}$. Let's simplify each fraction:
$\frac{96}{54}=\frac{16}{9}$? No, $96\div 6 = 16$, $54\div6 = 9$? Wait, $96\div 12 = 8$, $54\div12 = 4.5$? Wait, actually, $\frac{96}{54}=\frac{16}{9}\approx1.777$, $\frac{64}{45}\approx1.422$, that can't be. Wait, maybe the correspondence is different. Let's re - assign:
Suppose $CE$ corresponds to $MN$ (36), $CD$ corresponds to $LN$ (45), $DE$ corresponds to $LM$ (54). No, that doesn't work. Wait, the handwritten ratio is $\frac{96}{54}=\frac{64}{45}=\frac{80}{36}$. Let's simplify each:
$\frac{96}{54}=\frac{16}{9}\approx1.777$, $\frac{64}{45}\approx1.422$, $\frac{80}{36}=\frac{20}{9}\approx2.222$. Wait, that's not equal. Wait, maybe a typo in the handwritten part. Wait, maybe the correct correspondence is:
$\triangle CDE$: sides 64, 80, 96.
$\triangle LMN$: sides 36, 45, 54.

Now, $\frac{64}{36}=\frac{16}{9}$, $\frac{80}{45}=\frac{16}{9}$, $\frac{96}{54}=\frac{16}{9}$. Ah, there we go! So the corresponding sides are $CD = 64$ and $MN = 36$, $DE = 80$ and $LN = 45$, $CE = 96$ and $LM = 54$.

So $\frac{CD}{MN}=\frac{64}{36}=\frac{16}{9}$, $\frac{DE}{LN}=\frac{80}{45}=\frac{16}{9}$, $\frac{CE}{LM}=\frac{96}{54}=\frac{16}{9}$.

Since all three pairs of corresponding sides are in proportion ($\frac{16}{9}$), by the SSS (Side - Side - Side) similarity criterion, the two triangles $\triangle CDE$ and $\triangle MNL$ (or with correct correspondence) are similar.

Answer:

The triangles are similar by SSS similarity criterion.