Question

given: y || z
prove: m∠5 + m∠2 + m∠6 = 180°
angles lines statements reasons
def. of ≅ def. of straight angle
angle addition postulate alternate interior angles theorem
statements reasons
✓ 1. y || z 1. given

1. ∠1 ≅ ∠5 2. alternate interior angles theorem

correct! assemble the next statement
(diagram: line y with points l, a, m; line z with points c, b; triangle a-c-b with angles 1,2,3 at a; 4,5 at c; 6,7 at b)

Explanation:

Step1: Find the second alternate interior angle pair

Since \( y \parallel z \), by the alternate interior angles theorem, \( \angle 3 \cong \angle 6 \).

Step2: Use angle addition postulate for straight angle

The straight angle at \( A \) (on line \( LM \)) gives \( m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ \) (def. of straight angle).

Step3: Substitute congruent angles

Substitute \( \angle 1 \cong \angle 5 \) and \( \angle 3 \cong \angle 6 \) into the equation: \( m\angle 5 + m\angle 2 + m\angle 6 = 180^\circ \).

Filling the table (next statements and reasons):

• Statement 3: \( \angle 3 \cong \angle 6 \)

Reason 3: alternate interior angles theorem

• Statement 4: \( m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ \)

Reason 4: def. of straight angle

• Statement 5: \( m\angle 5 + m\angle 2 + m\angle 6 = 180^\circ \)

Reason 5: substitution (using \( \angle 1 \cong \angle 5 \) and \( \angle 3 \cong \angle 6 \))

Answer:

To complete the proof:

• Next Statement: \( \boldsymbol{\angle 3 \cong \angle 6} \)
• Next Reason: alternate interior angles theorem

(Subsequent steps follow with angle addition for the straight angle and substitution, as explained.)