Question

given: y || z prove: m∠5 + m∠2 + m∠6 = 180° angles lines statements reasons def. of ≅ substitution def. of straight angle angle addition postulate statements reasons 1. y || z 1. given 2. ∠1 ≅ ∠5 2. alternate interior angles theorem 3. ∠3 ≅ ∠6 3. alternate interior angles theorem 4. m∠1 = m∠5 4. def. of ≅ correct! assemble the next statement.

Explanation:

Step1: Recall angle addition postulate

Since \( \angle 1 \), \( \angle 2 \), and \( \angle 3 \) form a straight angle at point \( A \) on line \( y \), by the definition of a straight angle, \( m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ \).

Step2: Substitute \( m\angle 1 \) and \( m\angle 3 \)

From step 4, we know \( m\angle 1 = m\angle 5 \) and from the similar reasoning (or step 3 and def. of \( \cong \)) \( m\angle 3 = m\angle 6 \). Using substitution, replace \( m\angle 1 \) with \( m\angle 5 \) and \( m\angle 3 \) with \( m\angle 6 \) in the equation \( m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ \). So we get \( m\angle 5 + m\angle 2 + m\angle 6 = 180^\circ \).

Breaking down the proof steps in the table:

• Statement 5: \( m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ \)
• Reason 5: def. of straight angle (because \( \angle 1 \), \( \angle 2 \), \( \angle 3 \) form a straight angle on line \( y \))
• Statement 6: \( m\angle 5 + m\angle 2 + m\angle 6 = 180^\circ \)
• Reason 6: substitution (substituting \( m\angle 1 = m\angle 5 \) and \( m\angle 3 = m\angle 6 \) into the equation from statement 5)

Answer:

To complete the proof:

Statements	Reasons
6. \( m\angle 5 + m\angle 2 + m\angle 6 = 180^\circ \)	6. substitution

(If only the next statement after step 4 is needed, the next statement is \( m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ \) with reason "def. of straight angle", followed by the final statement with substitution.)