1. two right triangles are similar with side lengths of one triangle being 5, 12, and 13. if the shortest side of the second triangle is 10, what is the length of the longest of the second triangle?
a. 28
b. 40
c. 20
d. 30
2. the length of a buildings shadow is 20 meters at the same time that a 2 - meter stick casts a shadow of 4 meters. using similar triangles, what is the height of the building?
a. 15 meters
b. 10 meters
c. 5 meters
d. 20 meters
3. what does the aa criterion stand for in triangle similarity?
a. altitude - angle
b. area - angle
c. angle - area
d. angle - angle
4. a tree casts a shadow of 12 feet, while a 4 - foot stick casts a shadow of 6 feet. what is the height of the tree?
a. 12 feet
b. 6 feet
c. 8 feet
d. 10 feet
5. what is the criterion that determines triangle similarity if two pairs of angles are congruent?
a. aa criterion
b. sss criterion
c. sas criterion
d. right triangle criterion
6. in $ \triangle ghi, gh = 12, hi = 16 $ and $ \angle g = 80 ^ { \circ } $. in $ \triangle jkl, jk = 9, kl = 12 $ and $ \angle j = 80 ^ { \circ } $. are these triangles similar, and by which criterion?
a. no, the sides are not proportional
b. no, the angles are not equal
c. yes, by the aa criterion
d. yes, by the sas criterion
7. if $ \triangle abc \sim \triangle def, ab = 16, ac = 12 $ and $ de = 4 $, what is the length of $ df $?
a. 12
b. 9.5
c. 30
d. 9

Question

1. two right triangles are similar with side lengths of one triangle being 5, 12, and 13. if the shortest side of the second triangle is 10, what is the length of the longest of the second triangle?
   a. 28
   b. 40
   c. 20
   d. 30
2. the length of a buildings shadow is 20 meters at the same time that a 2 - meter stick casts a shadow of 4 meters. using similar triangles, what is the height of the building?
   a. 15 meters
   b. 10 meters
   c. 5 meters
   d. 20 meters
3. what does the aa criterion stand for in triangle similarity?
   a. altitude - angle
   b. area - angle
   c. angle - area
   d. angle - angle
4. a tree casts a shadow of 12 feet, while a 4 - foot stick casts a shadow of 6 feet. what is the height of the tree?
   a. 12 feet
   b. 6 feet
   c. 8 feet
   d. 10 feet
5. what is the criterion that determines triangle similarity if two pairs of angles are congruent?
   a. aa criterion
   b. sss criterion
   c. sas criterion
   d. right triangle criterion
6. in $ \triangle ghi, gh = 12, hi = 16 $ and $ \angle g = 80 ^ { \circ } $. in $ \triangle jkl, jk = 9, kl = 12 $ and $ \angle j = 80 ^ { \circ } $. are these triangles similar, and by which criterion?
   a. no, the sides are not proportional
   b. no, the angles are not equal
   c. yes, by the aa criterion
   d. yes, by the sas criterion
7. if $ \triangle abc \sim \triangle def, ab = 16, ac = 12 $ and $ de = 4 $, what is the length of $ df $?
   a. 12
   b. 9.5
   c. 30
   d. 9

Accepted Answer

# Explanation:
## Step1: Determine the ratio of similarity
The first triangle has side lengths $5$, $12$, and $13$. The shortest side of the first triangle is $5$, and the shortest side of the second triangle is $10$. The ratio of similarity $k=\frac{10}{5} = 2$.
## Step2: Find the longest side of the second triangle
The longest side of the first triangle is $13$. Using the ratio of similarity, the longest side of the second triangle $L=13\times k$. Substituting $k = 2$, we get $L=13\times2=26$.

# Answer:
a. $26$

# Explanation:
## Step1: Set up the proportion for similar triangles
Let $h$ be the height of the building. For similar triangles, $\frac{h}{20}=\frac{2}{4}$.
## Step2: Solve the proportion for $h$
Cross - multiply: $4h=2\times20$. Then $4h = 40$, and $h=\frac{40}{4}=10$.

# Answer:
b. $10$ meters

# Brief Explanations:
The AA (Angle - Angle) criterion for triangle similarity states that if two angles of one triangle are equal to two angles of another triangle, the triangles are similar.

# Answer:
d. Angle - Angle

# Explanation:
## Step1: Set up the proportion for similar triangles
Let $h$ be the height of the tree. Using the proportion $\frac{h}{12}=\frac{4}{6}$.
## Step2: Solve the proportion for $h$
Cross - multiply: $6h=4\times12$. Then $6h = 48$, and $h=\frac{48}{6}=8$.

# Answer:
c. $8$ feet

# Brief Explanations:
The AA (Angle - Angle) criterion is used to determine triangle similarity when two pairs of angles are congruent. For triangle congruence, when two pairs of angles are congruent (and a side in some cases for congruence, but if we just consider the angle - based criterion for congruence in the context of the question's options), the AA criterion (in the sense of AAS or ASA which are congruence criteria related to angles) is relevant. But strictly for the question "What is the criterion that determines triangle congruity if two pairs of angles are congruent",  the ASA (Angle - Side - Angle) or AAS (Angle - Angle - Side) are congruence criteria. However, if we assume the options are mis - labeled (since AA is similarity, but if we consider the closest in the given options), there is an error. But if we consider the standard, when two angles are congruent and a non - included side (AAS) or included side (ASA) is considered for congruence. But among the given options (assuming a typo in the question where it might mean similarity), but if we go by the options:
- SSS (Side - Side - Side) is for all three sides.
- SAS (Side - Angle - Side) is for two sides and included angle.
- Right Triangle criterion (like HL) is for right triangles.
- AA is similarity. But if we assume the question means congruence and there is a mis - option, but if we consider the closest in the given options (maybe a mis - print where AA is intended as AAS/ASA in the options), but strictly the answer based on the options:
# Answer:
a. AA Criterion (Note: There is a confusion as AA is similarity, but if we assume in the context of the given options and a possible mis - labeling in the question)

# Explanation:
## Step1: Check the ratio of sides and angles
For $\triangle GHI$ and $\triangle JKL$, $\frac{GH}{JK}=\frac{12}{9}=\frac{4}{3}$, $\frac{HI}{KL}=\frac{16}{12}=\frac{4}{3}$, and $\angle G=\angle J = 80^{\circ}$.
## Step2: Apply the SAS (Side - Angle - Side) similarity criterion
Since the ratio of two pairs of sides is equal ($\frac{GH}{JK}=\frac{HI}{KL}$) and the included angles ($\angle G$ and $\angle J$) are equal, by the SAS (Side - Angle - Side) similarity criterion, the triangles are similar.

# Answer:
d. Yes, by the SAS Criterion

# Explanation:
## Step1: Set up the proportion for similar triangles
Since $\triangle ABC\sim\triangle DEF$, $\frac{AB}{DE}=\frac{AC}{DF}$. Let $DF = x$, then $\frac{16}{4}=\frac{12}{x}$.
## Step2: Solve the proportion for $x$
Cross - multiply: $16x=4\times12$. Then $16x = 48$, and $x=\frac{48}{16}=3$.

# Answer:
d. $3$

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QUESTION IMAGE

Question

Explanation:

Step1: Determine the ratio of similarity

Step2: Find the longest side of the second triangle

Step1: Set up the proportion for similar triangles

Step2: Solve the proportion for \(h\)

Step1: Set up the proportion for similar triangles

Step2: Solve the proportion for \(h\)

Answer: