Question

given: y || z
prove: m∠5 + m∠2 + m∠6 = 180°
assemble the proof by dragging tiles to
the statements and reasons columns.
angles lines statements reasons
≅ = ∠1 ∠3 ∠5 ∠6
m∠1 m∠3 m∠5 m∠6
statements reasons

Explanation:

Step1: Identify Alternate Interior Angles

Since \( y \parallel z \), by the Alternate Interior Angles Theorem, \( \angle 1 \cong \angle 5 \) and \( \angle 3 \cong \angle 6 \). So \( m\angle 1 = m\angle 5 \) and \( m\angle 3 = m\angle 6 \).

Step2: Use Linear Pair Postulate

\( \angle 1 \), \( \angle 2 \), and \( \angle 3 \) form a linear pair, so \( m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ \) (Linear Pair Postulate: angles on a straight line sum to \( 180^\circ \)).

Step3: Substitute Angle Measures

Substitute \( m\angle 1 \) with \( m\angle 5 \) and \( m\angle 3 \) with \( m\angle 6 \) in the equation \( m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ \). This gives \( m\angle 5 + m\angle 2 + m\angle 6 = 180^\circ \).

Answer:

The proof is completed by showing \( \angle 1 \cong \angle 5 \), \( \angle 3 \cong \angle 6 \) (Alternate Interior Angles Theorem), using the Linear Pair Postulate for \( \angle 1, \angle 2, \angle 3 \), and substituting to get \( m\angle 5 + m\angle 2 + m\angle 6 = 180^\circ \).