Question

complete the proof that ( mangle rsv + mangle svu = 180^{circ} ).
1 ( overleftrightarrow{rt}paralleloverleftrightarrow{uw} ) given
2 ( angle svucongangle qsr )
3 ( mangle qsr + mangle rsv = 180^{circ} )
4 ( mangle rsv + mangle svu = 180^{circ} )

Explanation:

Step1: Corresponding angles postulate

When two parallel lines ($\overleftrightarrow{RT}$ and $\overleftrightarrow{UW}$) are cut by a transversal ($\overleftrightarrow{QX}$), corresponding angles are congruent. So, $\angle SVU\cong\angle QSR$ because of the Corresponding Angles Postulate.

Step2: Linear pair definition

$\angle QSR$ and $\angle RSV$ form a linear pair. By the definition of a linear pair, the sum of the measures of angles in a linear pair is $180^{\circ}$. So, $m\angle QSR + m\angle RSV=180^{\circ}$.

Step3: Substitution property

Since $\angle SVU\cong\angle QSR$ (from Step 1, so $m\angle SVU = m\angle QSR$), we can substitute $m\angle QSR$ with $m\angle SVU$ in the equation $m\angle QSR + m\angle RSV = 180^{\circ}$. Then $m\angle RSV + m\angle SVU=180^{\circ}$ by the Substitution Property.

Answer:

1. Corresponding Angles Postulate; 3. Definition of a linear pair; 4. Substitution Property.