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. $f - e = f - e$; slope from $p$ to $q = \frac{f}{e}$; $\frac{f}{e} = \frac{f}{e}$; definition of slope; slope cannot be a ratio. statement: drag and drop an item here; reason: definition of slope; slope from $q$ to $r = \frac{f}{e}$; reason: drag and drop an item here

Explanation:

Step1: Identify the first statement-reason pair

The reason "Definition of slope" should correspond to the statement that defines the slope between two points. For the slope from \( P \) to \( Q \), the slope is \( \frac{F}{E} \) by the definition of slope (rise over run). So the statement "Slope from \( P \) to \( Q=\frac{F}{E} \)" pairs with the reason "Definition of slope".

Step2: Use similarity of triangles

Since triangles 1 and 2 are similar, their corresponding sides are proportional. So \( \frac{F'}{E'}=\frac{F}{E} \) (corresponding sides of similar triangles are proportional).

Step3: Relate slopes

The slope from \( Q \) to \( R \) is \( \frac{F'}{E'} \) (by definition of slope, rise over run for triangle 1). And since \( \frac{F'}{E'}=\frac{F}{E} \) (from similar triangles), the slope from \( Q \) to \( R \) is equal to the slope from \( P \) to \( Q \), proving the slope is the same between any two points.

Answer:

• For the first (statement) with reason "Definition of slope": Statement is "Slope from \( P \) to \( Q=\frac{F}{E} \)".
• For the slope from \( Q \) to \( R=\frac{F'}{E'} \), the reason can be "Corresponding sides of similar triangles are proportional" (or using the proportion \( \frac{F'}{E'}=\frac{F}{E} \) from similar triangles). But based on the given options, the key pairings are:
• Statement: "Slope from \( P \) to \( Q=\frac{F}{E} \)" with Reason: "Definition of slope"
• Statement: "Slope from \( Q \) to \( R=\frac{F'}{E'} \)" with Reason (from similar triangles) related to \( \frac{F'}{E'}=\frac{F}{E} \) (the option \( \frac{F'}{E'}=\frac{F}{E} \) is the proportionality from similar triangles).

So the completed proof has:

• Statement: Slope from \( P \) to \( Q=\frac{F}{E} \); Reason: Definition of slope
• Statement: \( \frac{F'}{E'}=\frac{F}{E} \); Reason: Corresponding sides of similar triangles are proportional (or the given option \( \frac{F'}{E'}=\frac{F}{E} \))
• Statement: Slope from \( Q \) to \( R=\frac{F'}{E'} \); Reason: Definition of slope (or using the proportionality to show it equals the other slope)

But following the drag - and - drop options:

1. The statement paired with "Definition of slope" is "Slope from \( P \) to \( Q=\frac{F}{E} \)".
2. The statement "Slope from \( Q \) to \( R=\frac{F'}{E'} \)" can be justified by the definition of slope (rise over run for triangle 1), and the proportionality \( \frac{F'}{E'}=\frac{F}{E} \) (from similar triangles) shows it equals the slope from \( P \) to \( Q \).