Question

concepts and connections (27.0821000) yl ic (spring)
proving lines parallel
assignment active
writing a two-column proof
given: $angle t cong angle v$; $overline{st} parallel overline{uv}$
prove: $overline{tu} parallel overline{vw}$

statements | reasons
--- | ---

1. $overline{st} parallel overline{uv}$ | 1. given
2. $angle t cong angle u$ | 2. ♦
3. $angle t cong angle v$ | 3. given
4. $angle u cong angle v$ | 4. ♦
5. $overline{tu} parallel overline{vw}$ | 5. ♦

complete the two-column proof.
♦ =
♦ =
♦ =

Explanation:

Step1: Identify reason for ∠T≅∠U

$\overline{ST} \parallel \overline{UV}$, so alternate interior angles formed by transversal $\overline{TU}$ are congruent. This is the Alternate Interior Angles Theorem.

Step2: Identify reason for ∠U≅∠V

We know $\angle T \cong \angle U$ and $\angle T \cong \angle V$. If two angles are congruent to the same angle, they are congruent to each other (Transitive Property of Congruence).

Step3: Identify reason for $\overline{TU} \parallel \overline{VW}$

$\angle U \cong \angle V$ are alternate interior angles formed by transversal $\overline{UV}$ with lines $\overline{TU}$ and $\overline{VW}$. If alternate interior angles are congruent, the lines are parallel (Converse of Alternate Interior Angles Theorem).

Answer:

1. Alternate Interior Angles Theorem
2. Transitive Property of Congruence
3. Converse of Alternate Interior Angles Theorem