Question

ethan is proving that the slope between any two points on a straight line is the same
he has already proved that triangles 1 and 2 are similar
drag statements and reasons to complete the proof
definition of slope
$f - e = f - e$
slope cannot be a ratio
$\frac{f}{e} = \frac{f}{e}$
slope from $p$ to $q = \frac{f}{e}$
statement
reason
drag and drop
an item here
definition of slope
slope from $q$ to $r = \frac{f}{e}$
drag and drop
an item here
drag and drop
an item here
triangle 1 is similar to triangle 2

Explanation:

Step1: Analyze the first row (Reason: Definition of slope)

The definition of slope is the ratio of the vertical change (rise) to the horizontal change (run). For the slope from \( P \) to \( Q \), the rise is \( F \) and the run is \( E \), so the statement should be "Slope from \( P \) to \( Q = \frac{F}{E} \)".

Step2: Analyze the second row (Statement: Slope from \( Q \) to \( R = \frac{F'}{E'} \))

Since triangles 1 and 2 are similar, their corresponding sides are proportional. The reason for the slope from \( Q \) to \( R \) being \( \frac{F'}{E'} \) is related to the definition of slope, but wait, no—wait, the triangles are similar, so the ratios of corresponding sides are equal. Wait, the first row's reason is definition of slope, so the first statement (top row, statement) is "Slope from \( P \) to \( Q = \frac{F}{E} \)" (because definition of slope gives rise over run, which is \( F \) over \( E \) for \( P \) to \( Q \)). Then the second row: the statement is "Slope from \( Q \) to \( R = \frac{F'}{E'} \)", and the reason should be... Wait, no, the third row's reason is "Triangle 1 is similar to triangle 2", so the third row's statement should be the proportion from similar triangles, which is \( \frac{F}{E} = \frac{F'}{E'} \) (because in similar triangles, corresponding sides are proportional). Then the second row: the reason for "Slope from \( Q \) to \( R = \frac{F'}{E'} \)" is the definition of slope? Wait, no, let's re - order:

1. First row (Statement column, Reason: Definition of slope): The slope from \( P \) to \( Q \) is calculated using the definition of slope, which is rise over run. For \( P \) to \( Q \), rise is \( F \), run is \( E \), so statement is "Slope from \( P \) to \( Q = \frac{F}{E} \)".

2. Second row (Statement: Slope from \( Q \) to \( R = \frac{F'}{E'} \)): The reason for this is also the definition of slope? Wait, no, the third row's reason is "Triangle 1 is similar to triangle 2", so the third row's statement should be the proportion \( \frac{F}{E}=\frac{F'}{E'} \) (because in similar triangles, the ratios of corresponding sides are equal, and these sides represent the rise and run for the two slopes).

3. Third row (Reason: Triangle 1 is similar to triangle 2): The statement here is \( \frac{F}{E}=\frac{F'}{E'} \) (since similar triangles have proportional corresponding sides, and these sides are the rise and run for the slopes of \( PQ \) and \( QR \)).

So let's fill in the table:

- First row (Statement): Slope from \( P \) to \( Q = \frac{F}{E} \) (Reason: Definition of slope)
- Second row (Statement): Slope from \( Q \) to \( R = \frac{F'}{E'} \) (Reason: Definition of slope? Wait, no, maybe I messed up. Wait, the options given: the statements are "Definition of slope", " \( F - E=F' - E' \)" (no, that's not right), "Slope cannot be a ratio" (wrong), " \( \frac{F}{E}=\frac{F'}{E'} \)", "Slope from \( P \) to \( Q=\frac{F}{E} \)".

So let's match:

- First row (Reason: Definition of slope) → Statement: "Slope from \( P \) to \( Q=\frac{F}{E} \)" (because definition of slope is rise over run, so for \( P \) to \( Q \), rise is \( F \), run is \( E \))
- Second row (Statement: "Slope from \( Q \) to \( R=\frac{F'}{E'} \)") → Reason: "Definition of slope" (because for \( Q \) to \( R \), rise is \( F' \), run is \( E' \), so by definition of slope, slope is \( \frac{F'}{E'} \))
- Third row (Reason: "Triangle 1 is similar to triangle 2") → Statement: " \( \frac{F}{E}=\frac{F'}{E'} \)" (because in similar triangles, the ratios of corresponding sides (rise over run for each triangle) are equal)

So the correct drag - and - drop:

- First row (Statement): "Slope from \( P \) to \( Q=\frac{F}{E} \)" (Reason: Definition of slope)
- Second row (Statement: "Slope from \( Q \) to \( R=\frac{F'}{E'} \)") (Reason: Definition of slope)
- Third row (Statement: " \( \frac{F}{E}=\frac{F'}{E'} \)") (Reason: Triangle 1 is similar to triangle 2)

Answer:

• First row (Statement): Slope from \( P \) to \( Q=\frac{F}{E} \) (Reason: Definition of slope)
• Second row (Statement): Slope from \( Q \) to \( R=\frac{F'}{E'} \) (Reason: Definition of slope)
• Third row (Statement): \( \frac{F}{E}=\frac{F'}{E'} \) (Reason: Triangle 1 is similar to triangle 2)