Question

use the given information to complete the proof of the following theorem. if a line parallel to one side of a triangle intersects the other two sides, then it divides them proportionally. given: $overline{sr}paralleloverline{vu}$. prove: $\frac{sv}{vt}=\frac{ru}{ut}$. statement reason 1. $overline{sr}paralleloverline{vu}$ given 2. $angle tsrcongangle tvu$ if lines $parallel$, then corr. $angle scong$ 3. $angle tcongangle t$ reflexive property 4. $\triangle rstsim\triangle uvt$ reason 1 5. $\frac{st}{vt}=\frac{square}{ut}$ corr. sides of similar triangles are proportional 6. $st = sv+vt$ segment - addition property 7. $rt = ru + ut$ segment - addition property 8. $\frac{sv + vt}{vt}=\frac{square+square}{ut}$ substitution property 9. $\frac{sv}{vt}+\frac{vt}{vt}=\frac{ru}{ut}+\frac{ut}{ut}$ fraction algebra 10. $\frac{sv}{vt}+1=\frac{ru}{ut}+1$ fraction algebra 11. $\frac{sv}{vt}=\frac{square}{ut}$ addition and subtraction properties

Explanation:

Step1: Identify corresponding angles

Since $\overline{SR}\parallel\overline{VU}$, by the property of parallel - lines (if two lines are parallel, then corresponding angles are congruent), $\angle TSR\cong\angle TVU$.

Step2: Use reflexive property

$\angle T\cong\angle T$ by the reflexive property of congruence.

Step3: Prove similar triangles

By the AA (Angle - Angle) similarity criterion, $\triangle RST\sim\triangle UVT$ (because we have two pairs of congruent angles: $\angle TSR\cong\angle TVU$ and $\angle T\cong\angle T$).

Step4: Set up proportion of sides

For similar triangles $\triangle RST$ and $\triangle UVT$, the corresponding sides are proportional. So, $\frac{ST}{VT}=\frac{RT}{UT}$.

Step5: Apply segment - addition property

We know that $ST = SV+VT$ and $RT = RU + UT$ by the segment - addition property.

Step6: Substitute segment expressions

Substituting $ST = SV + VT$ and $RT = RU+UT$ into $\frac{ST}{VT}=\frac{RT}{UT}$, we get $\frac{SV + VT}{VT}=\frac{RU + UT}{UT}$.

Step7: Expand the fractions

Using fraction algebra, $\frac{SV + VT}{VT}=\frac{SV}{VT}+\frac{VT}{VT}$ and $\frac{RU + UT}{UT}=\frac{RU}{UT}+\frac{UT}{UT}$. So, $\frac{SV}{VT}+1=\frac{RU}{UT}+1$.

Step8: Solve for the desired proportion

By the addition and subtraction properties of equality, subtracting 1 from both sides of the equation $\frac{SV}{VT}+1=\frac{RU}{UT}+1$, we get $\frac{SV}{VT}=\frac{RU}{UT}$.

Answer:

The proof is completed as shown above to prove that if $\overline{SR}\parallel\overline{VU}$, then $\frac{SV}{VT}=\frac{RU}{UT}$.