Question

given: $\frac{hk}{ki}=\frac{jl}{li}$
prove: $overline{hj}paralleloverline{kl}$
statement reason

1. $\frac{hk}{ki}=\frac{jl}{li}$ given
2. $\frac{hk}{ki}+1=\frac{jl}{li}+1$ addition and subtraction properties
3. $\frac{hk}{ki}+\frac{ki}{ki}=\frac{jl}{li}+\frac{li}{li}$ fraction algebra
4. $\frac{hk + ki}{ki}=\frac{jl + li}{li}$ fraction algebra
5. $hi = hk+square$ segment addition property
6. $ji = jl+square$ segment addition property
7. $\frac{hi}{ki}=\frac{ji}{li}$ substitution property
8. $angle icongangle i$ reflexive property
9. $\triangle hijsim\trianglesquare$ sas similarity property
10. $angle ihjcongangle ikl$ corr. $angle$s of similar triangles are $cong$
11. $overline{hj}parallelsquare$ if corr. $angle$s $cong$ then lines $parallel$

Explanation:

Step1: Given ratio

$\frac{HK}{KI}=\frac{JL}{LI}$

Step2: Add 1 to both sides

$\frac{HK}{KI}+1=\frac{JL}{LI}+1$

Step3: Rewrite 1 as a fraction

$\frac{HK}{KI}+\frac{KI}{KI}=\frac{JL}{LI}+\frac{LI}{LI}$

Step4: Combine fractions

$\frac{HK + KI}{KI}=\frac{JL+LI}{LI}$

Step5: Segment - addition property

$HI = HK+KI$ and $JI=JL + LI$

Step6: Substitute

$\frac{HI}{KI}=\frac{JI}{LI}$

Step7: Reflexive property of angles

$\angle I\cong\angle I$

Step8: SAS similarity

$\triangle HIJ\sim\triangle KIL$ (by Side - Angle - Side (SAS) similarity since $\frac{HI}{KI}=\frac{JI}{LI}$ and $\angle I$ is common)

Step9: Corresponding angles of similar triangles

$\angle IHJ\cong\angle IKL$ (corresponding angles of similar triangles are congruent)

Step10: Parallel lines

$\overline{HJ}\parallel\overline{KL}$ (if corresponding angles are congruent, then the lines are parallel)

Answer:

$\overline{HJ}\parallel\overline{KL}$