Select the correct answer.

**Given**: r∥s

**Prove**: mr=ms

StatementsReasons

1. r∥s

given

2. mr=d−bc−0=d−bc

ms=0−ac−0=-ac

application of the slope formula

3. distance from (0,b) to (0,a) equals the distance from (c,d) to (c,0)

definition of parallel lines

4. ?

application of the distance formula

5. mr=(b−a)−bc

substitution property of equality

6. mr=-ac

inverse property of addition

7. mr=ms

substitution property of equality

The table and corresponding image show the proof of the relationship between the slopes of two parallel lines. What is the missing statement in step 4?


 **A.** 

b−c=d−a

 **B.** 

c−0=a−b

 **C.** 

d−0=b−a

 **D.** 

c−d=b−a


To determine the correct answer for the missing statement in step 4, we need to understand the application of the distance formula and how it relates to the slopes of the parallel lines \( r \) and \( s \).

### Step 3 Recap:
In step 3, it is stated that the distance from \( (0,b) \) to \( (0,a) \) equals the distance from \( (c,d) \) to \( (c,0) \). This implies that the vertical distances between the points on each line are equal.

### Step 4: Application of the Distance Formula
The distance formula is:
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Since we are only dealing with vertical distances, the formula simplifies to the absolute difference between the \( y \)-coordinates:
\[
|b - a| = |d - 0|
\]
This simplifies to:
\[
b - a = d - 0
\]
or
\[
b - a = d
\]

Thus, the correct missing statement in step 4 should be:
\[
\text{d − 0 = b − a}
\]

### Final Answer:
The correct choice is:

**C.** \( d − 0 = b − a \)

Would you like to dive deeper into any specific step or have additional questions? Here are some related queries:

1. How does the distance formula help in proving parallel lines?
2. What is the significance of equal distances in proving line slopes?
3. Can two non-parallel lines have the same slope?
4. How does the definition of parallel lines relate to their slopes?
5. What is the substitution property of equality?

**Tip:** When solving geometric proofs, carefully follow the logical progression from one statement to the next to ensure the validity of each step.

Explore the proof that the slopes of two parallel lines r and s are equal. Understand the application of the distance formula and the properties of parallel lines in geometry. Suitable for high school students studying geometry and algebra.

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