Question

complete the proof. given: ∠gnd and ∠pdk are supplementary. prove: line gr is parallel to line fk. statement: ∠gnd and ∠pdk are supplementary; reason: given. statement: m∠gnd + m∠pdk = 180°; reason: definition of supplementary angles. statement: ∠gnd ≅ ∠bnr; reason: blank. statement: m∠gnd = m∠bnr; reason: definition of congruence. statement: m∠bnr + m∠pdk = 180°; reason: substitution. statement: ∠bnr and ∠pdk are supplementary; reason: blank. statement: line gr is parallel to line fk; reason: blank

Explanation:

Step1: Identify vertical - angle property

$\angle GND$ and $\angle BNR$ are vertical angles. Vertical angles are congruent. So the reason for $\angle GND\cong\angle BNR$ is "Vertical Angles Theorem".

Step2: Recall supplementary - angle definition

If the sum of the measures of two angles is $180^{\circ}$, they are supplementary. Since $m\angle BNR + m\angle PDK=180^{\circ}$, by the definition of supplementary angles, $\angle BNR$ and $\angle PDK$ are supplementary.

Step3: Apply parallel - line theorem

If two lines are cut by a transversal such that a pair of same - side interior angles are supplementary, then the two lines are parallel. $\angle BNR$ and $\angle PDK$ are same - side interior angles for lines $GR$ and $FK$ with transversal $BP$. So the reason for "Line $GR$ is parallel to line $FK$" is "Converse of Same - Side Interior Angles Postulate".

Answer:

1. Reason for $\angle GND\cong\angle BNR$: Vertical Angles Theorem
2. Reason for $\angle BNR$ and $\angle PDK$ are supplementary: Definition of Supplementary Angles
3. Reason for Line $GR$ is parallel to line $FK$: Converse of Same - Side Interior Angles Postulate