Question

complete the proof that $m\angle tsv = m\angle r + m\angle v$ without using the triangle angle sum theorem.
(image of a geometric figure with points u, t, s, v, r and a table with statements and reasons)
1 $\overline{su} \parallel \overline{rv}$
2 $\angle r \cong \angle tsu$
3 $\angle v \cong \angle usv$
4 $m\angle tsv = m\angle tsu + m\angle usv$
5 $m\angle tsv = m\angle r + m\angle v$

Explanation:

Step1: Identify Parallel Lines Reason

From the diagram, \(\overline{SU}\) and \(\overline{RV}\) have the same direction (marked with red arrows), so they are parallel by the definition of parallel lines (or corresponding arrow markings indicating parallelism). So for statement 1, the reason is "Given (or by the diagram's parallel markings)".

Step2: Corresponding Angles Reason

Since \(\overline{SU} \parallel \overline{RV}\) and \(\overline{RS}\) is a transversal, \(\angle R\) and \(\angle TSU\) are corresponding angles. By the Corresponding Angles Theorem (if two parallel lines are cut by a transversal, corresponding angles are congruent), \(\angle R \cong \angle TSU\). So reason for statement 2: "Corresponding Angles Theorem".

Step3: Alternate Interior Angles Reason

\(\overline{SU} \parallel \overline{RV}\) and \(\overline{VS}\) is a transversal, so \(\angle V\) and \(\angle USV\) are alternate interior angles. By the Alternate Interior Angles Theorem (if two parallel lines are cut by a transversal, alternate interior angles are congruent), \(\angle V \cong \angle USV\). Reason for statement 3: "Alternate Interior Angles Theorem".

Step4: Angle Addition Postulate

The Angle Addition Postulate states that if a point lies in the interior of an angle, the sum of the two smaller angles equals the larger angle. Here, \(S\) is a point such that \(\angle TSV\) is composed of \(\angle TSU\) and \(\angle USV\), so \(m\angle TSV = m\angle TSU + m\angle USV\) by the Angle Addition Postulate. Reason for statement 4: "Angle Addition Postulate".

Step5: Substitution Property

From statements 2 and 3, we know \(m\angle R = m\angle TSU\) and \(m\angle V = m\angle USV\) (since congruent angles have equal measures). Substituting these into the equation from statement 4, we get \(m\angle TSV = m\angle R + m\angle V\). Reason for statement 5: "Substitution Property (using statements 2, 3, and 4)".

Answer:

1. Reason: Given (or by diagram's parallel markings)
2. Reason: Corresponding Angles Theorem
3. Reason: Alternate Interior Angles Theorem
4. Reason: Angle Addition Postulate
5. Reason: Substitution Property (using congruent angle measures from 2, 3 and 4)