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 (checkmark) 1. ( y parallel z ) 1. given 2. ( angle 1 cong angle 5 ) 2. alternate interior angles theorem (checkmark) 3. ( angle 3 cong angle 6 ) 3. alternate interior angles theorem

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 (which measures \(180^\circ\)), we have \(m\angle1 + m\angle2 + m\angle3 = 180^\circ\) (by the definition of a straight angle, since \(LAM\) is a straight line, the sum of angles \(\angle1\), \(\angle2\), and \(\angle3\) on this straight line is \(180^\circ\)).

Step2: Use substitution from congruent angles

We know from steps 2 and 3 that \(\angle1\cong\angle5\) (so \(m\angle1 = m\angle5\)) and \(\angle3\cong\angle6\) (so \(m\angle3 = m\angle6\)). 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 the congruent angles' measures) and also based on the angle addition postulate and straight angle definition. But for the proof table:

Statement 4:

\(m\angle1 + m\angle2 + m\angle3 = 180^\circ\) (Reason: def. of straight angle, because \(LAM\) is a straight line, so the sum of angles on a straight line is \(180^\circ\))

Statement 5:

\(m\angle5 + m\angle2 + m\angle6 = 180^\circ\) (Reason: substitution, using \(m\angle1 = m\angle5\) and \(m\angle3 = m\angle6\) from steps 2 and 3 into the equation from statement 4)

But looking at the given table, after steps 2 and 3, we need to connect to the sum. Let's structure it properly:

First, from the straight line \(y\) (line \(LAM\)), the angles \(\angle1\), \(\angle2\), \(\angle3\) form a straight angle, so \(m\angle1 + m\angle2 + m\angle3 = 180^\circ\) (def. of straight angle). Then, since \(\angle1\cong\angle5\) and \(\angle3\cong\angle6\), we can substitute \(m\angle1\) with \(m\angle5\) and \(m\angle3\) with \(m\angle6\), so \(m\angle5 + m\angle2 + m\angle6 = 180^\circ\) (substitution).

So in the proof table, the next statement (after 3) would be:

1. \(m\angle1 + m\angle2 + m\angle3 = 180^\circ\)

Reason: def. of straight angle

1. \(m\angle5 + m\angle2 + m\angle6 = 180^\circ\)

Reason: substitution (using \(\angle1\cong\angle5\) and \(\angle3\cong\angle6\) to replace \(m\angle1\) with \(m\angle5\) and \(m\angle3\) with \(m\angle6\) in statement 4)

Answer:

To complete the proof:

Statement 4:

\(m\angle1 + m\angle2 + m\angle3 = 180^\circ\)

Reason 4:

def. of straight angle (since \(LAM\) is a straight line, the sum of angles on it is \(180^\circ\))

Statement 5:

\(m\angle5 + m\angle2 + m\angle6 = 180^\circ\)

Reason 5:

substitution (replace \(m\angle1\) with \(m\angle5\) and \(m\angle3\) with \(m\angle6\) using \(\angle1\cong\angle5\) and \(\angle3\cong\angle6\) from steps 2–3)

(If only one statement/reason is needed to fill the box, the key statement is \(m\angle5 + m\angle2 + m\angle6 = 180^\circ\) with reason "substitution" (after establishing \(m\angle1 + m\angle2 + m\angle3 = 180^\circ\) via straight angle definition).)