Question

completing a proof

given: △stu with \\(\overline{st} \parallel \overline{xy}\\)
prove: \\(\frac{sx}{xu} = \frac{ty}{yu}\\)

complete the steps of the proof.

statements	reasons
2. ∠1 and ∠2 are corresponding angles	2. def. of corresponding angles
3. ∠3 and ∠4 are corresponding angles	3. def. of corresponding angles
4. ∠1 ≅ ∠2; ∠3 ≅ ∠4	4. corresponding angles theorem
5. ⋆	5. aa similarity theorem
6. \\(\frac{su}{xu} = \frac{tu}{yu}\\)	6. def. of similar triangles
7. \\(su = sx + xu\\); \\(tu = ty + yu\\)	7. segment addition postulate
8. \\(\frac{sx + xu}{xu} = \frac{ty + yu}{yu}\\)	8. substitution property
9. \\(\frac{sx}{xu} + \frac{xu}{xu} = \frac{ty}{yu} + \frac{yu}{yu}\\)	9. addition of fractions
10. \\(\frac{sx}{xu} + 1 = \frac{ty}{yu} + 1\\)	10. simplification
11. \\(\frac{sx}{xu} = \frac{ty}{yu}\\)	11. ⋆

options for step 5: angle stu is congruent to angle xyu; triangle stu is congruent to triangle xyu; triangle stu is similar to triangle xyu

Explanation:

Step1: Recall AA Similarity

AA (Angle - Angle) similarity theorem states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. We know from steps 2 - 4 that $\angle1\cong\angle2$ and $\angle3\cong\angle4$. So, in $\triangle STU$ and $\triangle XYU$, we have two pairs of congruent angles.

Step2: Apply AA Similarity

For $\triangle STU$ and $\triangle XYU$, $\angle1\cong\angle2$ (corresponding angles) and $\angle3\cong\angle4$ (corresponding angles). By the AA similarity theorem, $\triangle STU\sim\triangle XYU$. So the statement for step 5 should be "triangle STU is similar to triangle XYU".

Step11: Recall Proportionality from Similar Triangles

From step 6, we have $\frac{SU}{XU}=\frac{TU}{YU}$. From step 7, $SU = SX+XU$ and $TU=TY + YU$. After substitution (step 8), addition of fractions (step 9), and simplification (step 10), we get $\frac{SX}{XU}+ 1=\frac{TY}{YU}+1$. Subtracting 1 from both sides (subtraction property of equality), we get $\frac{SX}{XU}=\frac{TY}{YU}$. So the reason for step 11 is "subtraction property of equality" (or equivalent, like "subtracting 1 from both sides"). But for step 5, the correct statement is about the similarity of the triangles.

Answer:

For step 5: triangle STU is similar to triangle XYU

For step 11: subtraction property of equality (or equivalent, e.g., "subtract 1 from both sides")