Question

prove: $overline{gx}paralleloverline{hk}$ an incomplete two - column proof shown includes some statements and reasons to prove $overline{gx}paralleloverline{hk}$. complete each missing statement. statements reasons 1. $overline{yp}perpoverline{gx}$ and $overline{yp}perpoverline{hk}$ 1. given 2. $angle gsy$ is a right angle, $angle hnp$ is a right angle 2. definition of perpendicular lines 3. $mangle gsy = 90^{circ},mangle hnp = 90^{circ}$ 3. definition of right angle 4. $mangle gsy + mangle hnp=90^{circ}+90^{circ}$ 4. 5. $mangle gsy + mangle hnp = 180^{circ}$ 5. addition property of equality 6. $angle gsy$ and $angle hnp$ are supplementary 6. definition of supplementary angles 7. $overline{gx}paralleloverline{hk}$ 7.

Explanation:

Step1: Substitute angle measures

Since $m\angle GSY = 90^{\circ}$ and $m\angle HNP=90^{\circ}$, we substitute these values into the left - hand side and right - hand side of the equation for step 4. So the reason for step 4 is "Substitution property of equality".

Step2: Use parallel line criterion

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. $\angle GSY$ and $\angle HNP$ are same - side interior angles. So the reason for step 7 is "If two lines are cut by a transversal so that same - side interior angles are supplementary, then the lines are parallel".

Answer:

1. Substitution property of equality
2. If two lines are cut by a transversal so that same - side interior angles are supplementary, then the lines are parallel