Question

complete the proof that m∠uvx + m∠qsr = 180°. statement reason 1 uw || rt given 2 ∠uvx ≅ ∠rsv 3 m∠qsr + m∠rsv = 180° 4 m∠uvx + m∠qsr = 180°

Explanation:

Step1: Identify angle - pair relationship

Since $\overleftrightarrow{UW}\parallel\overleftrightarrow{RT}$, $\angle UVX$ and $\angle RSV$ are alternate - interior angles. By the Alternate Interior Angles Theorem, if two parallel lines are cut by a transversal, then the alternate interior angles are congruent. So, $\angle UVX\cong\angle RSV$ because of the Alternate Interior Angles Theorem.

Step2: Identify linear - pair relationship

$\angle QSR$ and $\angle RSV$ form a linear pair. By the Linear Pair Postulate, if two angles form a linear pair, then the sum of their measures is $180^{\circ}$. So, $m\angle QSR + m\angle RSV=180^{\circ}$ because of the Linear Pair Postulate.

Step3: Substitute equal angles

Since $\angle UVX\cong\angle RSV$, then $m\angle UVX = m\angle RSV$. Substituting $m\angle UVX$ for $m\angle RSV$ in the equation $m\angle QSR + m\angle RSV = 180^{\circ}$, we get $m\angle UVX + m\angle QSR=180^{\circ}$ by the Substitution Property of Equality.

Answer:

1. Reason: Alternate Interior Angles Theorem
2. Reason: Linear Pair Postulate
3. Reason: Substitution Property of Equality