Question

given: y || z prove: m∠5 + m∠2 + m∠6 = 180° diagram: line y (lm) with point a, line z (cb) with points c, b. triangle abc with angles 1 (∠lac), 2 (∠cab), 3 (∠bam), 4 (∠lca), 5 (∠acb), 6 (∠cba), 7 (∠mba). tiles: angles, lines, statements, reasons (||, =, ∠1, ∠3, ∠5, ∠6, m∠1, m∠3, m∠5, m∠6). instructions: assemble the proof by dragging tiles to statements and reasons columns.

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

Answer:

The proof is assembled by using the Alternate Interior Angles Theorem to relate \( \angle 1 \) to \( \angle 5 \) and \( \angle 3 \) to \( \angle 6 \), then applying the Linear Pair Postulate to \( \angle 1 \), \( \angle 2 \), \( \angle 3 \), and substituting the angle measures to show \( m\angle 5 + m\angle 2 + m\angle 6 = 180^\circ \).