Question

given: ∠rns and ∠sng are a linear - pair, ∠sng and ∠gnb are a linear - pair. prove: ∠rns≅∠gnb. statements: 1. ∠rns and ∠sng are a linear pair. 2. ∠rns and ∠sng are supplementary. 3. ∠sng and ∠gnb are supplementary. 4. ∠rns≅∠gnb. reasons: 1. given 2. linear pair postulate 3. 4. congruent supplements theorem. choose the correct reason for step 3.

Explanation:

Step1: Recall linear - pair definition

If two angles form a linear pair, they are supplementary. Given that $\angle RNS$ and $\angle SNG$ are a linear pair, then $\angle RNS+\angle SNG = 180^{\circ}$. Also, since $\angle SNG$ and $\angle GNB$ are a linear pair, $\angle SNG+\angle GNB=180^{\circ}$.

Step2: Use the substitution property

We know that $\angle RNS+\angle SNG=\angle SNG+\angle GNB = 180^{\circ}$. By the subtraction property of equality, if we subtract $\angle SNG$ from both sides of the equation $\angle RNS+\angle SNG=\angle SNG+\angle GNB$, we get $\angle RNS=\angle GNB$.

Step3: Recall congruent - angle definition

Since $\angle RNS$ and $\angle GNB$ have equal measures, by the definition of congruent angles (angles with equal measures are congruent), we can say that $\angle RNS\cong\angle GNB$.

Answer:

The proof is completed as above to show that $\angle RNS\cong\angle GNB$ using the linear - pair postulate, substitution property, and the definition of congruent angles.