Question

in exercises 11-16, $overline{xj} cong overline{jy}$, $overline{yl} cong overline{lz}$, and $overline{xk} cong overline{kz}$. complete the statement. example 4

1. $overline{jk} \parallel$ ___
2. $overline{jl} \parallel$ ___
3. $overline{xy} \parallel$ ___
4. $overline{jy} \cong$ _ $\cong$ _
5. $overline{jl} \cong$ _ $\cong$ _
6. $overline{jk} \cong$ _ $\cong$ _

Explanation:

Step1: Apply Midsegment Theorem

Given $\overline{XJ}\cong\overline{JY}$, $\overline{YL}\cong\overline{LZ}$, $\overline{XK}\cong\overline{KZ}$, so $J,K,L$ are midpoints.

Step2: Solve 11: Identify parallel segment

$\overline{JK}$ connects midpoints of $\overline{XY}$ and $\overline{XZ}$, so $\overline{JK}\parallel\overline{YZ}$

Step3: Solve 12: Identify parallel segment

$\overline{JL}$ connects midpoints of $\overline{XY}$ and $\overline{YZ}$, so $\overline{JL}\parallel\overline{XZ}$

Step4: Solve 13: Identify parallel segment

$\overline{XY}$ corresponds to segment through midpoints of $\overline{XZ}$ and $\overline{YZ}$, so $\overline{XY}\parallel\overline{KL}$

Step5: Solve 14: Identify congruent segments

$\overline{JY}\cong\overline{XJ}$, and $\overline{XJ}\cong\overline{KL}$ (midsegment = $\frac{1}{2}$ of side, $\overline{JY}=\frac{1}{2}\overline{XY}=\overline{KL}$), so $\overline{JY}\cong\overline{XJ}\cong\overline{KL}$

Step6: Solve 15: Identify congruent segments

$\overline{JL}$ is midsegment, $\overline{JL}=\frac{1}{2}\overline{XZ}=\overline{XK}=\overline{KZ}$, so $\overline{JL}\cong\overline{XK}\cong\overline{KZ}$

Step7: Solve 16: Identify congruent segments

$\overline{JK}$ is midsegment, $\overline{JK}=\frac{1}{2}\overline{YZ}=\overline{YL}=\overline{LZ}$, so $\overline{JK}\cong\overline{YL}\cong\overline{LZ}$

Answer:

1. $\overline{YZ}$
2. $\overline{XZ}$
3. $\overline{KL}$
4. $\overline{XJ}$, $\overline{KL}$
5. $\overline{XK}$, $\overline{KZ}$
6. $\overline{YL}$, $\overline{LZ}$