Question

uw || rt. complete the proof that m∠qsr + m∠uvx = 180°.
statement:
1 uw || rt
2 ∠uvx ≅ ∠rsv
3 m∠qsr + m∠rsv = 180°
4 m∠qsr + m∠uvx = 180°
reason: (blank for each statement)

Explanation:

Step1: Identify Given Information

We know \(\overleftrightarrow{UW} \parallel \overleftrightarrow{RT}\) (given).

Step2: Determine Angle Relationship (Statement 2)

Since \(\overleftrightarrow{UW} \parallel \overleftrightarrow{RT}\) and a transversal cuts them, \(\angle UVX\) and \(\angle RSV\) are corresponding angles. So, \(\angle UVX \cong \angle RSV\) by the Corresponding Angles Postulate (when two parallel lines are cut by a transversal, corresponding angles are congruent).

Step3: Linear Pair or Supplementary Angles (Statement 3)

\(\angle QSR\) and \(\angle RSV\) form a linear pair (they are adjacent and form a straight line), so \(m\angle QSR + m\angle RSV = 180^\circ\) by the Linear Pair Postulate (linear pair of angles are supplementary).

Step4: Substitution Property (Statement 4)

Since \(\angle UVX \cong \angle RSV\), we can substitute \(m\angle UVX\) for \(m\angle RSV\) in the equation \(m\angle QSR + m\angle RSV = 180^\circ\). Thus, \(m\angle QSR + m\angle UVX = 180^\circ\) by the Substitution Property of Equality.

Answer:

Statement	Reason
2. \(\angle UVX \cong \angle RSV\)	Corresponding Angles Postulate (Parallel lines cut by transversal, corresponding angles congruent)
3. \(m\angle QSR + m\angle RSV = 180^\circ\)	Linear Pair Postulate (Linear pair angles are supplementary)
4. \(m\angle QSR + m\angle UVX = 180^\circ\)	Substitution Property of Equality (Substitute \(m\angle UVX\) for \(m\angle RSV\) since \(\angle UVX \cong \angle RSV\))