Question

given: $\frac{sv}{vt}=\frac{ru}{ut}$
prove: $overline{sr}paralleloverline{vu}$

statement | reason

1. $\frac{sv}{vt}=\frac{ru}{ut}$ | given
2. $\frac{sv}{vt}+1=\frac{ru}{ut}+1$ | reason?
3. $\frac{sv}{vt}+\frac{vt}{vt}=\frac{ru}{ut}+\frac{ut}{ut}$ | fraction algebra
4. $\frac{sv + vt}{vt}=\frac{ru + ut}{ut}$ | fraction algebra
5. $st = sv+vt$ | reason?
6. $rt = ru + ut$ | reason?
7. $\frac{st}{vt}=\frac{?}{ut}$ | substitution property
8. $angle tcongangle?$ | reflexive property
9. $\triangle rstsim\triangle uvt$ | reason?
10. $angle tsrcongangle?$ | corr. $angle$s of similar triangles are $cong$
11. $overline{sr}paralleloverline{?}$ | reason?

Explanation:

Step1: Add 1 to both sides

$\frac{SV}{VT}+1=\frac{RU}{UT}+1$

Step2: Rewrite 1 as a fraction

$\frac{SV}{VT}+\frac{VT}{VT}=\frac{RU}{UT}+\frac{UT}{UT}$

Step3: Combine fractions

$\frac{SV + VT}{VT}=\frac{RU + UT}{UT}$

Step4: Segment - addition postulate

$ST=SV + VT$ and $RT=RU + UT$

Step5: Substitute

$\frac{ST}{VT}=\frac{RT}{UT}$

Step6: Reflexive property

$\angle T\cong\angle T$

Step7: SAS similarity criterion

$\triangle RST\sim\triangle UVT$ (Side - Angle - Side similarity, since $\frac{ST}{VT}=\frac{RT}{UT}$ and $\angle T$ is common)

Step8: Corresponding angles of similar triangles

$\angle TSR\cong\angle TVU$

Step9: Converse of corresponding - angles postulate

$\overline{SR}\parallel\overline{VU}$ (If corresponding angles are congruent, then the lines are parallel)

Answer:

The proof is completed as above to show $\overline{SR}\parallel\overline{VU}$