Question

$$ \overline { v y } \cong \overline { v x }, \angle v x z \cong \angle v y w $$, and $$ \angle w v x \cong \angle y v z $$. complete the proof that $$ \overline { v w } \cong \overline { v z } $$. 1 $$ \overline { v y } \cong \overline { v x } $$ given 2 $$ \angle v x z \cong \angle v y w $$ given 3 $$ \angle w v x \cong \angle y v z $$ given 4 $$ m \angle x v z = m \angle x v y + m \angle y v z $$ additive property of angle measure 5 $$ m \angle w v y = m \angle w v x + m \angle x v y $$ additive property of angle measure 6 $$ m \angle x v z = m \angle x v y + m \angle w v x $$ substitution 7 $$ m \angle w v y = m \angle x v z $$ 8 $$ \delta v x z \cong \delta v y w $$ asa 9 $$ \overline { v w } \cong \overline { v z } $$ cpctc

Explanation:

Step1: Analyze the equations

From step 4: \(m\angle XVZ=m\angle XVY + m\angle YVZ\), step 5: \(m\angle W VY=m\angle WVX + m\angle XVY\), and step 3: \(\angle WVX\cong\angle YVZ\) (so \(m\angle WVX = m\angle YVZ\)).

Step2: Substitute in step 5

Substitute \(m\angle YVZ\) for \(m\angle WVX\) in the equation \(m\angle W VY=m\angle WVX + m\angle XVY\). Then \(m\angle W VY=m\angle YVZ + m\angle XVY\).

Step3: Compare with step 4

Since \(m\angle XVZ=m\angle XVY + m\angle YVZ\) (step 4) and \(m\angle W VY=m\angle YVZ + m\angle XVY\) (after substitution in step 5), by the commutative property of addition (\(a + b=b + a\)), we can say \(m\angle WVY=m\angle XVZ\).

Answer:

Commutative Property of Addition.