1. **Problem 1: Decide if triangles ABC and DEF are similar.** Given side lengths: - Triangle ABC: $AB=4$, $BC=8$, $AC=10$ - Triangle DEF: $FD=4$, $DE=2$, $FE=5$ 2. **Similarity criteria:** Triangles are similar if their corresponding angles are equal or their corresponding sides are proportional. 3. **Check side ratios:** $$\frac{AB}{FD} = \frac{4}{4} = 1$$ $$\frac{BC}{DE} = \frac{8}{2} = 4$$ $$\frac{AC}{FE} = \frac{10}{5} = 2$$ Since the ratios are not equal, the triangles are **not similar**. 4. **Problem 2: Decide if triangles MNX and XYZ are similar.** Given only angle markings at $N$ and $Z$ (both marked), no side lengths. 5. **Similarity criteria:** If two angles of one triangle are equal to two angles of another triangle, the triangles are similar by AA (Angle-Angle) similarity. 6. Since angles at $N$ and $Z$ are marked and presumably equal, triangles $MNX$ and $XYZ$ are similar by AA. Similarity statement: $\triangle MNX \sim \triangle XYZ$ by AA similarity. 7. **Problem 3: Decide if triangles STU and PQR are similar.** Given side lengths: - Triangle STU: $TS=21$, $SU=9$, angle at $S$ marked - Triangle PQR: $PR=6$, $QR=14$, angle at $R$ marked 8. **Check side ratios for corresponding sides:** Assuming $TS$ corresponds to $PR$ and $SU$ corresponds to $QR$: $$\frac{TS}{PR} = \frac{21}{6} = 3.5$$ $$\frac{SU}{QR} = \frac{9}{14} \approx 0.64$$ Ratios are not equal, so sides are not proportional. 9. **Check angles:** Angles at $S$ and $R$ are marked, but only one pair of angles is known. 10. **Conclusion:** Not enough information to confirm similarity; based on side ratios, triangles are **not similar or not necessarily similar**. **Final answers:** - Triangles ABC and DEF: Not similar or not necessarily similar. - Triangles MNX and XYZ: Similar, $\triangle MNX \sim \triangle XYZ$ by AA similarity. - Triangles STU and PQR: Not similar or not necessarily similar.