A. The graph shows four points on the line and the triangles resulting from sketching on the rise and run from one point to the next. Viewing the graph or R-value as a transversal to the rise and run segments, are the two pairs of corresponding angles along the transversal congruent? Explain.
B. What is the relationship between the two resulting triangles? Explain your answer.
C. Use the relationship from Part B to state the relationship between corresponding sides of the resulting triangles.
D. Use the properties of similar triangles to explain why the slope, or rise-to-run ratio, is constant between any two points on the line.
E. Use any two points to find the unit rate for this proportional relationship. Show your work.

A. From the given graph, given that the angles are formed between the transversal and lines parallel to the R-value axis and the transversal and lines parallel to the thickness axis, the angle formed by the transversal and each parallel lines are corresponding angles and are therefore congruent

The tan of one of the angles on each triangle are

On the uppermost triangle, tanθ₁ = (9.42 - 3.14)/(3 - 1) = 3.14

On the other triangle, tanθ₂ =(25.12 - 12.56)/(8 - 4) = 3.14

tanθ₁ = tanθ₂

∴ θ₁ = θ₂

ΔABC and ΔDEF are right triangles, therefore, all the angles of ΔABC and ΔDEF are congruent

B. Given that the interior angles of the triangles formed by the line the graph are congruent, the triangles are similar by AA similarity postulate

C. Given that the triangles are similar, the length of the corresponding sides of the triangles are proportional

D. Given that the slope is the tangent of the angle between the line of the graph and the Thickness axis, and that the angle is the same for both triangles, (∠θ₁ and ∠θ₂) the slope or the rise-to-run ratio is constant for the two triangles and also for any two points on the line

E. The Unit of proportionality is found using any two points as follows;

Using points used to show the proportional relationship on triangle ΔABC are (8, 25.12) and (4, 12.56) and on triangle ΔDEF, the two points used to show proportional relationship are (3, 9.42), and (1, 3.14)

The ratio of the corresponding sides on ΔABC and ΔDEF are

[tex]\overline {AC}[/tex]/[tex]\overline {DF}[/tex] = (25.12 - 12.56)/(9.42 - 3.14) = 2

[tex]\overline {BC}[/tex]/[tex]\overline {EF}[/tex] = (8 - 4)/(3 - 1) = 2

Therefore, the unit rate of proportionality which is the scale factor = 2.