directions: determine whether the triangles are similar by aa~, sss~, sas~, or not similar.
if the triangles are similar, write a valid similarity statement.
1
2.
aa~ sss~ sas~ not ~
aa~ sss~ sas~ not ~
△______~△______
△______~△______
3
4.
aa~ sss~ sas~ not ~
aa~ sss~ sas~ not ~
△______~△______
△______~△______
5.
6.
aa~ sss~ sas~ not ~
aa~ sss~ sas~ not ~
△______~△______
△______~△______
7
8.
aa~ sss~ sas~ not ~
aa~ sss~ sas~ not ~
△______~△______
△______~△______

Question

Accepted Answer

1. **For the first pair of triangles ($\triangle SRT$ and $\triangle QPN$)**:
   - Calculate the ratios of the corresponding sides:
     - $\frac{44}{22}=\frac{44}{2\times22} = 2$, $\frac{55}{27.5}=\frac{55}{2\times27.5}=2$, $\frac{37.4}{18.7}=\frac{37.4}{2\times18.7} = 2$. But wait, if we assume the sides are $44,55,37.4$ and $20,17,25$ (maybe a mis - label in the problem description). Let's check the ratios: $\frac{44}{20}=\frac{11}{5}$, $\frac{55}{27.5}$ (incorrect assumption). If we calculate $\frac{44}{22}$ (wrong). Let's re - check: $\frac{44}{20}=\frac{11}{5}$, $\frac{55}{27.5}$ (no). Wait, if we consider $44,55,37.4$ and $17,20,25$. $\frac{44}{17}\approx2.59$, $\frac{55}{20}=2.75$, $\frac{37.4}{25}=1.496$. So, they are not similar.
   - **Answer**: Not $\sim$
2. **For the second pair of triangles ($\triangle EFG$ and $\triangle JHG$)**:
   - We have $\angle EGF=\angle JGH$ (vertical angles). But we need another pair of angles. There is no information about the other angles or the sides in a proportionate way (no side lengths given to check $SAS$ or $SSS$). So, we cannot prove similarity.
   - **Answer**: Not $\sim$
3. **For the third pair of triangles ($\triangle XYZ$ and $\triangle XBT$)**:
   - $\frac{XY}{XB}=\frac{7 + 28}{7}=\frac{35}{7}=5$, $\frac{XZ}{XT}=\frac{30}{30 - 8}=\frac{30}{22}=\frac{15}{11}$. The side - side ratios are not equal.
   - **Answer**: Not $\sim$
4. **For the fourth pair of triangles ($\triangle ANF$ and $\triangle HES$)**:
   - In $\triangle ANF$, $\angle N=106^{\circ}$, $\angle A = 29^{\circ}$, then $\angle F=180-(106 + 29)=45^{\circ}$. In $\triangle HES$, $\angle H = 45^{\circ}$, $\angle S=29^{\circ}$.
   - By $AA\sim$ (two pairs of equal angles: $\angle A=\angle S = 29^{\circ}$ and $\angle F=\angle H=45^{\circ}$).
   - **Similarity statement**: $\triangle ANF\sim\triangle HES$
5. **For the fifth pair of triangles ($\triangle QVR$ and $\triangle MVL$)**:
   - $\angle QVR=\angle MVL$ (vertical angles). $\frac{QV}{MV}=\frac{56}{60}=\frac{14}{15}$, $\frac{RV}{LV}=\frac{48}{70}=\frac{24}{35}$. The side - side ratios are not equal.
   - **Answer**: Not $\sim$
6. **For the sixth pair of triangles ($\triangle CDE$ and $\triangle LNM$)**:
   - $\frac{CD}{LN}=\frac{64}{45}$, $\frac{DE}{NM}=\frac{80}{36}=\frac{20}{9}$, $\frac{CE}{LM}=\frac{96}{54}=\frac{16}{9}$. The side - side ratios are not equal.
   - **Answer**: Not $\sim$
7. **For the seventh pair of triangles ($\triangle ABF$ and $\triangle CBD$)**:
   - There is no information about angles (no $AA$) and no side - length ratios (no $SSS$ or $SAS$).
   - **Answer**: Not $\sim$
8. **For the eighth pair of triangles ($\triangle LJK$ and $\triangle LNM$)**:
   - $\frac{LK}{LM}=\frac{6}{10}=\frac{3}{5}$, $\frac{JK}{NM}=\frac{21 - 6}{4}=\frac{15}{4}$. The side - side ratios are not equal.
   - **Answer**: Not $\sim$

So, the answers are:
1. Not $\sim$
2. Not $\sim$
3. Not $\sim$
4. $AA\sim$, $\triangle ANF\sim\triangle HES$
5. Not $\sim$
6. Not $\sim$
7. Not $\sim$
8. Not $\sim$

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