Question

tv || sw. complete the proof that m∠utw = m∠s + m∠w without using the triangle angle sum theorem.

(image of a triangle and lines: w at the top, s at the bottom left, t at the bottom center, u to the right of t on a horizontal line, v above t with an arrow. then a table with statement and reason columns:

1. tv || sw, reason: given
2. ∠s ≅ ∠utv, reason: blank
3. ∠w ≅ ∠vtw, reason: alternate interior angles theorem
4. m∠utw = m∠utv + m∠vtw, reason: additive property of angle measure
5. m∠utw = m∠s + m∠w, reason: blank)

Explanation:

Step1: Identify Reason for Statement 2

Since \(\overline{TV} \parallel \overline{SW}\) and \(\angle S\) and \(\angle UTV\) are alternate interior angles (formed by transversal \(SU\) intersecting the parallel lines \(\overline{TV}\) and \(\overline{SW}\)), we use the Alternate Interior Angles Theorem. So the reason for \(\angle S \cong \angle UTV\) is the Alternate Interior Angles Theorem.

Step2: Identify Reason for Statement 5

From Statement 2, \(\angle S \cong \angle UTV\) implies \(m\angle S = m\angle UTV\) (Definition of Congruent Angles). From Statement 3, \(\angle W \cong \angle VTW\) implies \(m\angle W = m\angle VTW\) (Definition of Congruent Angles). Substitute these into Statement 4 (\(m\angle UTW = m\angle UTV + m\angle VTW\)) using the Substitution Property of Equality. So the reason for \(m\angle UTW = m\angle S + m\angle W\) is the Substitution Property (substituting \(m\angle S\) for \(m\angle UTV\) and \(m\angle W\) for \(m\angle VTW\)).

Answer:

• For Statement 2 Reason: Alternate Interior Angles Theorem
• For Statement 5 Reason: Substitution Property (or Substitution, since we substitute \(m\angle S\) for \(m\angle UTV\) and \(m\angle W\) for \(m\angle VTW\) into the equation from Statement 4)

(If filling the table:
Row 2 Reason: Alternate Interior Angles Theorem
Row 5 Reason: Substitution Property of Equality)