Allana knows that △XYZ∼△UVZ and XY⎯⎯⎯⎯⎯⎯⎯⎯∥UV⎯⎯⎯⎯⎯⎯⎯⎯⎯. What segment relationships can Allana use to prove that the slope of line l is equal to the slope of line m?
Allana can show that UVXY=VZYZ using the properties of similar triangles. Then, UVVZ=XYYZ by the Multiplicative Property of Equality. So, the slopes are equal by the slope formula.
Allana can show that UZXZ=VZYZ using the properties of similar triangles. Then, UZVZ=XZYZ by the Multiplicative Property of Equality. So, the slopes are equal by the slope formula.
Allana can show that UZXZ=UVXY using the properties of similar triangles. Then, UZXY=UVXZ by the Multiplicative Property of Equality. So, the slopes are equal by the slope formula.
Allana can show that UZXZ=VZYZ using the properties of similar triangles. Then, UZVZ=XZYZ by the Multiplicative Property of Equality. So, the slopes are equal by the slope formula.
The correct segment relationships that Allana can use to prove that the slope of line l is equal to the slope of line m are:
Allana can show that UZXZ=VZYZ using the properties of similar triangles. This is because the corresponding sides of similar triangles are proportional. In this case, the corresponding sides UZ and VZ, as well as XZ and YZ, are proportional.
Then, Allana can use the fact that UZVZ=XZYZ by the Multiplicative Property of Equality, which states that if two ratios are equal, their reciprocals are also equal. The ratios UZ/VZ and XZ/YZ are equal since they correspond to the sides of similar triangles, and by taking reciprocals, Allana arrives at the equality UZVZ=XZYZ.
Finally, Allana can conclude that the slopes of lines l and m are equal by applying the slope formula. The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula (y2 - y1) / (x2 - x1). Allana can substitute UZ for (x1, y1) and VZ for (x2, y2) to find the slope of line l, and substitute XZ for (x1, y1) and YZ for (x2, y2) to find the slope of line m. Since UZVZ=XZYZ, the slopes of lines l and m will be equal.