Question

triangle is 180°
given: $y \parallel z$
prove: $m\angle5 + m\angle2 + m\angle6 = 180^\circ$
angles lines statements reasons
$\cong$ $=$ $\angle1$ $\angle3$ $\angle5$ $\angle6$
$m\angle1$ $m\angle3$ $m\angle5$ $m\angle6$
statements reasons
1 $y \parallel z$ 1 given
hint

Explanation:

Step1: Identify alternate interior angles

$\angle 1 \cong \angle 5$, $\angle 3 \cong \angle 6$ (Alternate Interior Angles Theorem, since $y \parallel z$)

Step2: Use straight angle sum

$m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ$ (Straight angle definition, $\angle LAM$ is a straight angle)

Step3: Substitute congruent angles

Substitute $m\angle 1 = m\angle 5$ and $m\angle 3 = m\angle 6$ into the equation:
$m\angle 5 + m\angle 2 + m\angle 6 = 180^\circ$

Answer:

The completed two-column proof is:

Statements	Reasons
2. $\angle 1 \cong \angle 5$, $\angle 3 \cong \angle 6$	2. Alternate Interior Angles Theorem
3. $m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ$	3. Definition of a straight angle
4. $m\angle 5 + m\angle 2 + m\angle 6 = 180^\circ$	4. Substitution Property of Equality

Final proven result: $\boldsymbol{m\angle 5 + m\angle 2 + m\angle 6 = 180^\circ}$