Question

for exercises 1–4, if the two triangles are similar, state why they are similar. if not, state that they are not similar.
1.
2.
3.
4.
for exercises 5 and 6, use the triangles shown.

1. what is fe?
2. what is de?

for exercises 7 and 8, what is the value of x?
7.
8.

1. are triangles abc, def, and hij similar? explain.
2. the width of the pond shown is x ft. what is the value of x?

Explanation:

Step1: Verify similarity for Q1

Triangles $JKL$ and $MNO$: Only one pair of corresponding angles is marked equal, no side info. No similarity criteria met.

Step2: Verify similarity for Q2

Triangles $MLN$ and $QOP$: Ratios: $\frac{26}{13}=2$, $\frac{30}{15}=2$. But $MLN$ has two sides 26, $QOP$ has two sides 13: corresponding sides don't match (non-included angle not confirmed equal). No similarity.

Step3: Verify similarity for Q3

Triangles $ADB$ and $CEB$: Vertical angles at $B$ equal, one pair of sides marked equal, but no other equal angles/sides. No similarity criteria met.

Step4: Verify similarity for Q4

Triangles $TUV$ and $TWX$: Right angles equal, $\angle T$ is common. $\frac{TU}{TV}=\frac{14}{28}=\frac{1}{2}$, $\frac{TW}{TX}=\frac{5}{10}=\frac{1}{2}$. SAS similarity holds.

Step5: Find FE for Q5

Triangles $PQR$ and $EFD$ are similar (AA: $35^\circ, 89^\circ$ equal). Ratio: $\frac{FE}{PQ}=\frac{FD}{PR}$. $\frac{FE}{20}=\frac{24}{40}$ → $FE=20\times\frac{24}{40}=12$.

Step6: Find DE for Q6

Use similarity ratio $\frac{DE}{QR}=\frac{FD}{PR}$. $\frac{DE}{36}=\frac{24}{40}$ → $DE=36\times\frac{24}{40}=21.6$.

Step7: Find x for Q7

Triangles $ABC$ and $DFE$ are similar (AA: two pairs of angles equal). Ratio: $\frac{x}{AB}=\frac{EF}{BC}$. $\frac{x}{10}=\frac{12}{6}$ → $x=10\times2=20$.

Step8: Find x for Q8

Triangles are similar (AA: base angles equal). Ratio: $\frac{x}{14}=\frac{3}{21}$. $x=14\times\frac{3}{21}=2$.

Step9: Verify similarity for Q9

Right triangles:

• $ABC$: sides 3,4,5 → ratios $\frac{3}{5}=0.6$, $\frac{4}{5}=0.8$
• $DEF$: sides 6,8,10 → ratios $\frac{6}{10}=0.6$, $\frac{8}{10}=0.8$
• $HIJ$: sides 9,12,15 → ratios $\frac{9}{15}=0.6$, $\frac{12}{15}=0.8$

All sides are proportional (SSS similarity).

Step10: Find x for Q10

Triangles $ABE$ and $CDE$ are similar (AA: vertical angles, alternate interior angles equal). Ratio: $\frac{x}{60}=\frac{80}{48}$. $x=60\times\frac{80}{48}=100$.

Answer:

1. Not similar
2. Not similar
3. Not similar
4. Similar by SAS similarity (common angle, proportional sides)
5. $12$
6. $21.6$
7. $20$
8. $2$
9. Yes, all three are similar. They are right triangles with side lengths in the 3:4:5 ratio (SSS similarity, all corresponding sides are proportional).
10. $100$ ft