Question

1. given ( lm = 30 ), ( mn = 40 ), ( ln = 50 ), ( pq = 15 ), ( qr = 20 ), and ( pr = 25 ). prove ( \triangle lmn sim \triangle pqr )

proof:

statements	reason
2. ( \frac{lm}{pq} = \frac{mn}{qr} = \frac{ln}{pr} )	2.
3.	3. substitution
4.	4. simplify
5. ( \triangle lmn sim \triangle pqr )	5.

Explanation:

Step1: Fill in Statement 1

The given information is \( LM = 30 \), \( MN = 40 \), \( LN = 50 \), \( PQ = 15 \), \( QR = 20 \), and \( PR = 25 \). So Statement 1 should list these given lengths: \( LM = 30 \), \( MN = 40 \), \( LN = 50 \), \( PQ = 15 \), \( QR = 20 \), \( PR = 25 \)

Step2: Reason for Statement 2

To show the ratios of corresponding sides are equal, the reason for Statement 2 (setting up the proportion of corresponding sides) is "Corresponding Sides of Triangles (to check for similarity)" or more precisely, the idea that if we want to prove similarity by SSS (Side - Side - Side) similarity criterion, we first set up the ratios of corresponding sides. So the reason for Statement 2 is "To check for SSS similarity, set up ratios of corresponding sides"

Step3: Fill in Statement 3

Using substitution (substituting the given values into the ratio equation), we substitute \( LM = 30 \), \( MN = 40 \), \( LN = 50 \), \( PQ = 15 \), \( QR = 20 \), \( PR = 25 \) into \( \frac{LM}{PQ}=\frac{MN}{QR}=\frac{LN}{PR} \). So Statement 3 is \( \frac{30}{15}=\frac{40}{20}=\frac{50}{25} \)

Step4: Fill in Statement 4

Simplify each fraction: \( \frac{30}{15} = 2 \), \( \frac{40}{20}=2 \), \( \frac{50}{25} = 2 \). So Statement 4 is \( 2 = 2 = 2 \)

Step5: Reason for Statement 5

Since the ratios of all three corresponding sides of \( \triangle LMN \) and \( \triangle PQR \) are equal, by the SSS (Side - Side - Side) Similarity Criterion, which states that if the corresponding sides of two triangles are proportional, then the triangles are similar. So the reason for Statement 5 is "SSS (Side - Side - Side) Similarity Criterion"

Answer:

1. Statement 1: \( LM = 30 \), \( MN = 40 \), \( LN = 50 \), \( PQ = 15 \), \( QR = 20 \), \( PR = 25 \)
2. Reason for Statement 2: To check for SSS similarity, set up ratios of corresponding sides
3. Statement 3: \( \frac{30}{15}=\frac{40}{20}=\frac{50}{25} \)
4. Statement 4: \( 2 = 2 = 2 \)
5. Reason for Statement 5: SSS (Side - Side - Side) Similarity Criterion (If the corresponding sides of two triangles are proportional, the triangles are similar)