Question

for question #17, respond completely in your answer document.

1. complete the proof that shows how we can determine that $\overline{jn}\cong\overline{ko}$.

given: m is midpoint of $\overline{jk}$
$\overline{jn}\parallel\overline{ok}$
prove: prove $\overline{jn}\cong\overline{ko}$

1. m is the midpoint of $\overline{jk}$
2. $\overline{jm}\cong\overline{km}$
3. $\overline{jn}\parallel\overline{ok}$
4. $\angle n\cong\angle o$
5. $\angle jmn\cong\angle kmo$
6. $\triangle jmn\cong\triangle kmo$
7. $\overline{jn}\cong\overline{ko}$
8. given
9. ____________
10. given
11. ____________
12. ____________
13. ____________
14. ____________

Explanation:

Step1: Midpoint Definition

By the definition of a midpoint, if \(M\) is the midpoint of \(\overline{JK}\), then \(\overline{JM}\cong\overline{KM}\).

Step2: Alternate Interior Angles

Since \(\overline{JN}\parallel\overline{OK}\), by the Alternate Interior Angles Theorem, \(\angle N\cong\angle O\).

Step3: Vertical Angles

\(\angle JMN\) and \(\angle KMO\) are vertical angles. By the Vertical Angles Theorem, \(\angle JMN\cong\angle KMO\).

Step4: AAS Congruence

In \(\triangle JMN\) and \(\triangle KMO\), we have \(\angle N\cong\angle O\), \(\angle JMN\cong\angle KMO\), and \(\overline{JM}\cong\overline{KM}\). So, by the Angle - Angle - Side (AAS) congruence criterion, \(\triangle JMN\cong\triangle KMO\).

Step5: CPCTC

If \(\triangle JMN\cong\triangle KMO\), then by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem, \(\overline{JN}\cong\overline{KO}\).

Answer:

1. Definition of midpoint
2. Alternate Interior Angles Theorem
3. Vertical Angles Theorem
4. AAS (Angle - Angle - Side)
5. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)