in exercises 1 and 2, determine whether the triangles are similar. if they are, write a similarity statement. explain your reasoning.
1. triangles with right angles and angles 51° and 39°
2. triangles with angles 21°, 93° and 21°, 64°
in exercises 3 and 4, show that the two triangles are similar.
3. $\triangle abd$ and $\triangle ace$ triangle diagram with parallel lines
4. $\triangle wxz$ and $\triangle zxy$ triangle diagram with right angle and angles 66°, 24°
5. the height of the empire state building is 1250 feet tall. your friend, who is 6 feet 3 inches tall, is standing nearby and casts a shadow that is 33 inches long. what is the length of the shadow of the empire state building?

Question

Accepted Answer

### Problem 1: Determine if Triangles are Similar (First Pair)
# Explanation:
## Step1: Analyze Triangle $ \triangle ABC $
$ \triangle ABC $ is right - angled at $ A $, so $ \angle A = 90^{\circ} $, and $ \angle B=51^{\circ} $. Then $ \angle C=180^{\circ}-\angle A - \angle B=180 - 90 - 51 = 39^{\circ} $.
## Step2: Analyze Triangle $ \triangle NMI $
$ \triangle NMI $ is right - angled at $ M $, so $ \angle M = 90^{\circ} $, and $ \angle N = 39^{\circ} $. Then $ \angle I=180^{\circ}-\angle M-\angle N = 180 - 90 - 39=51^{\circ} $.
## Step3: Check for Similarity
In $ \triangle ABC $ and $ \triangle NMI $, $ \angle A=\angle M = 90^{\circ} $, $ \angle B=\angle I = 51^{\circ} $, $ \angle C=\angle N = 39^{\circ} $. By the AA (Angle - Angle) similarity criterion, the triangles are similar. The similarity statement is $ \triangle ABC\sim\triangle MIN $ (order of angles should match: $ \angle A=\angle M $, $ \angle B=\angle I $, $ \angle C=\angle N $).

### Problem 2: Determine if Triangles are Similar (Second Pair)
# Explanation:
## Step1: Analyze Triangle $ \triangle QFH $
In $ \triangle QFH $, we know two angles: $ \angle Q = 21^{\circ} $, $ \angle F=93^{\circ} $. Then $ \angle H=180-(21 + 93)=66^{\circ} $.
## Step2: Analyze Triangle $ \triangle SRT $
In $ \triangle SRT $, we know two angles: $ \angle S = 21^{\circ} $, $ \angle R = 64^{\circ} $. Then $ \angle T=180-(21 + 64)=95^{\circ} $.
## Step3: Check for Similarity
Since the angle measures of the two triangles are not the same (e.g., $ \angle F = 93^{\circ}$ and $ \angle T=95^{\circ}$, $ \angle H = 66^{\circ}$ and $ \angle R = 64^{\circ}$), the triangles are not similar.

### Problem 3: Show $ \triangle ABD\sim\triangle ACE $
# Explanation:
## Step1: Identify Corresponding Angles
$ \angle A$ is common to both $ \triangle ABD $ and $ \triangle ACE $. Also, since $ BD\parallel CE $ (from the arrows indicating parallel lines), $ \angle ABD=\angle ACE $ (corresponding angles) and $ \angle ADB=\angle AEC $ (corresponding angles).
## Step2: Apply AA Similarity
By the AA (Angle - Angle) similarity criterion, since $ \angle A=\angle A $ and $ \angle ABD=\angle ACE $ (or $ \angle ADB=\angle AEC $), $ \triangle ABD\sim\triangle ACE $.

### Problem 4: Show $ \triangle WXZ\sim\triangle ZXY $
# Explanation:
## Step1: Analyze Angles in $ \triangle WXZ $
In $ \triangle WXZ $, $ \angle WXZ = 90^{\circ} $, $ \angle W=66^{\circ} $, so $ \angle WZX=180-(90 + 66)=24^{\circ} $.
## Step2: Analyze Angles in $ \triangle ZXY $
In $ \triangle ZXY $, $ \angle XZY = 24^{\circ} $, $ \angle Y = 24^{\circ} $? Wait, no. Wait, $ \angle WXZ=\angle XZY = 90^{\circ} $? No, $ \angle WXZ = 90^{\circ} $, and $ \angle ZXY = 90^{\circ} $ (since $ ZX\perp WY $). Also, $ \angle WZX=\angle Y = 24^{\circ} $.
## Step3: Apply AA Similarity
$ \angle WXZ=\angle ZXY = 90^{\circ} $, and $ \angle WZX=\angle Y = 24^{\circ} $. By AA similarity criterion, $ \triangle WXZ\sim\triangle ZXY $.

### Problem 5: Shadow Length of Empire State Building
# Explanation:
## Step1: Convert Units
First, convert the height of the friend to inches. 6 feet 3 inches: 1 foot = 12 inches, so 6 feet $=6\times12 = 72$ inches, and 6 feet 3 inches $=72 + 3=75$ inches. Let the length of the shadow of the Empire State Building be $ x $ inches.
## Step2: Set up Proportion
Since the triangles formed by the friend and his shadow, and the Empire State Building and its shadow are similar (by AA similarity, as the sun's rays are parallel, so the angles of elevation are equal), we can set up the proportion: $ \frac{\text{Height of friend}}{\text{Length of friend's shadow}}=\frac{\text{Height of building}}{\text{Length of building's shadow}} $
So, $ \frac{75}{33}=\frac{1250\times12}{x} $ (we convert 1250 feet to inches: 1250 feet $ = 1250\times12$ inches).
## Step3: Solve for $ x $
Cross - multiply: $ 75x=33\times1250\times12 $
$ x=\frac{33\times1250\times12}{75} $
First, simplify $ \frac{33\times1250\times12}{75}=\frac{33\times1250\times12}{75}=33\times\frac{1250}{75}\times12 $
$ \frac{1250}{75}=\frac{50}{3} $
So $ x = 33\times\frac{50}{3}\times12=11\times50\times12 = 6600 $ inches.
Convert back to feet: $ \frac{6600}{12}=550 $ feet.

### Answers:
1. $ \triangle ABC\sim\triangle MIN $ (by AA similarity)
2. The triangles are not similar.
3. $ \triangle ABD\sim\triangle ACE $ (by AA similarity)
4. $ \triangle WXZ\sim\triangle ZXY $ (by AA similarity)
5. The length of the shadow of the Empire State Building is 550 feet.

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QUESTION IMAGE

Question

Explanation:

Problem 1: Determine if Triangles are Similar (First Pair)

Step1: Analyze Triangle \( \triangle ABC \)

Step2: Analyze Triangle \( \triangle NMI \)

Step3: Check for Similarity

Problem 2: Determine if Triangles are Similar (Second Pair)

Step1: Analyze Triangle \( \triangle QFH \)

Step2: Analyze Triangle \( \triangle SRT \)

Step3: Check for Similarity

Problem 3: Show \( \triangle ABD\sim\triangle ACE \)

Step1: Identify Corresponding Angles

Step2: Apply AA Similarity

Problem 4: Show \( \triangle WXZ\sim\triangle ZXY \)

Answer: