Question

given: ( y parallel z ) prove: ( mangle 5 + mangle 2 + mangle 6 = 180^circ ) angles lines statements reasons def of ( cong ) substitution def of straight angle angle addition postulate statements reasons 1. ( y parallel z ) 1. given 2. ( angle 1 cong angle 5 ) 2. alternate interior angles theorem 3. ( angle 3 cong angle 6 ) 3. alternate interior angles theorem 4. ( mangle 1 = mangle 5 ) 4. def of ( cong ) correct! assemble the next statement.

Explanation:

Step1: Recall angle addition postulate

The angle addition postulate states that if a point lies in the interior of an angle, the sum of the two adjacent angles formed is equal to the measure of the original angle. For a straight angle (like $\angle LAM$ which is $180^\circ$), we have $m\angle1 + m\angle2 + m\angle3 = 180^\circ$ (by the definition of a straight angle, since $\angle LAM$ is a straight line, its measure is $180^\circ$, and by angle addition postulate, the sum of $\angle1$, $\angle2$, and $\angle3$ is equal to $m\angle LAM$).

Step2: Substitute $m\angle1$ and $m\angle3$

From step 4, we know $m\angle1 = m\angle5$ and from the similar logic (since $\angle3\cong\angle6$, so $m\angle3 = m\angle6$ by definition of congruent angles). So we substitute $m\angle1$ with $m\angle5$ and $m\angle3$ with $m\angle6$ in the equation $m\angle1 + m\angle2 + m\angle3 = 180^\circ$.
So the next statement should be $m\angle5 + m\angle2 + m\angle6 = 180^\circ$ and the reason is substitution (since we substituted $m\angle1$ with $m\angle5$ and $m\angle3$ with $m\angle6$) and also based on the previous steps where we used angle addition postulate for the straight angle. But following the proof structure:

First, the statement before the last one (the one we need to fill) should be using the angle addition postulate on $\angle1$, $\angle2$, $\angle3$. So:

Step1: Apply angle addition postulate

Statement: $m\angle1 + m\angle2 + m\angle3 = 180^\circ$
Reason: def of straight angle (and angle addition postulate, since $\angle LAM$ is a straight angle, so $m\angle LAM = 180^\circ$, and by angle addition postulate, $m\angle1 + m\angle2 + m\angle3 = m\angle LAM$)

Step2: Substitute $m\angle1$ and $m\angle3$

Statement: $m\angle5 + m\angle2 + m\angle6 = 180^\circ$
Reason: substitution (using $m\angle1 = m\angle5$ and $m\angle3 = m\angle6$ from step 4 and the congruence of $\angle3$ and $\angle6$)

Answer:

The next statement (step 5) is $m\angle1 + m\angle2 + m\angle3 = 180^\circ$ with reason "def of straight angle (angle addition postulate)", and then step 6 is $m\angle5 + m\angle2 + m\angle6 = 180^\circ$ with reason "substitution". But following the immediate next step after step 4, the first missing statement is $m\angle1 + m\angle2 + m\angle3 = 180^\circ$ (reason: def of straight angle, angle addition postulate), then the final statement is $m\angle5 + m\angle2 + m\angle6 = 180^\circ$ (reason: substitution).