Question

1. whether the triangles are similar. if similar, state how (aa-, sss-, or sas-), and write a similarity statement. 2. 3. 4. 5. 6. 7. 8.

Explanation:

Step1: Recall triangle - similarity criteria

The main triangle - similarity criteria are AA (angle - angle), SSS (side - side - side), and SAS (side - angle - side).

Step2: Analyze problem 3

In problem 3, we have two triangles $\triangle ABE$ and $\triangle DBC$. In $\triangle ABE$, $\angle A = 53^{\circ}$ and $\angle ABE=42^{\circ}$, so $\angle AEB=180^{\circ}-(53^{\circ} + 42^{\circ})=85^{\circ}$. In $\triangle DBC$, $\angle DBC = 85^{\circ}$. Since $\angle AEB=\angle DBC$ and $\angle ABE$ and $\angle DBC$ are vertical angles, by AA similarity criterion, $\triangle ABE\sim\triangle DBC$.

Step3: Analyze problem 5

In problem 5, since $KM\parallel JN$, we have $\angle LKM=\angle LJN$ and $\angle LMK=\angle LNJ$ (corresponding angles). By AA similarity criterion, $\triangle LKM\sim\triangle LJN$.

Step4: Analyze problem 6

For problem 6, we calculate the ratios of the corresponding sides. $\frac{FT}{FD}=\frac{30}{30 + 35}=\frac{30}{65}=\frac{6}{13}$, $\frac{ST}{ED}=\frac{20}{20 + 24}=\frac{20}{44}=\frac{5}{11}$. Since the ratios of the corresponding sides are not equal, the two triangles are not similar.

Step5: Analyze problem 7

In problem 7, in $\triangle XYZ$, $\angle X = 61^{\circ}$ and $\angle Z = 90^{\circ}$, so $\angle Y=29^{\circ}$. In $\triangle GHJ$, $\angle G = 90^{\circ}$ and $\angle J = 34^{\circ}$, so $\angle H = 56^{\circ}$. Since the corresponding angles are not equal, the two triangles are not similar.

Step6: Analyze problem 8

For problem 8, we calculate the ratios of the corresponding sides. $\frac{QR}{RS}=\frac{52}{39}=\frac{4}{3}$, $\frac{TR}{RU}=\frac{63}{84}=\frac{3}{4}$. Since the ratios of the corresponding sides are not equal, the two triangles are not similar.

Answer:

1. Similar by AA; $\triangle ABE\sim\triangle DBC$
2. Similar by AA; $\triangle LKM\sim\triangle LJN$
3. Not similar
4. Not similar
5. Not similar