Question

problem 12: from this given statement, select the definition, property, postulate, or theorem that justifies the prove statement. given: ∠3 ≅ ∠5 prove: ℓ || m (diagram of lines ℓ, m, n with angles 1–8)

Explanation:

1. Identify the angles: $\angle 3$ and $\angle 5$ are alternate interior angles (formed by transversal $n$ intersecting lines $\ell$ and $m$).
2. Recall the theorem: The Alternate Interior Angles Theorem says that if alternate interior angles are congruent, the lines are parallel.
3. Apply the theorem: Given $\angle 3\cong\angle 5$ (alternate interior angles), so $\ell\parallel m$ by the Alternate Interior Angles Theorem.

Answer:

The theorem that justifies this is the "Alternate Interior Angles Theorem" (or Postulate), which states that if two lines are cut by a transversal and the alternate interior angles are congruent, then the two lines are parallel. Here, $\angle 3$ and $\angle 5$ are alternate interior angles formed by transversal $n$ cutting lines $\ell$ and $m$. Since $\angle 3\cong\angle 5$, by the Alternate Interior Angles Theorem, we can conclude that $\ell\parallel m$.