Question

line ( p ) is parallel to line ( q ). which proof explains why the sum of the interior angles of the triangle must be ( 180^{circ} )?
(a) ( angle 4 cong angle 2 ) and ( angle 7 cong angle 2 ) because they are alternate interior angles of parallel lines cut by a transversal.
( m angle 1 + m angle 2 + m angle 3 = 180^{circ} ) because they form a straight angle.
( m angle 4 + m angle 2 + m angle 7 = 180^{circ} ) by the substitution property of equality.
( m angle 5 + m angle 2 + m angle 6 = 180^{circ} ) by supplementary angles.
(b) ( angle 5 cong angle 1 ) and ( angle 6 cong angle 3 ) because they are alternate interior angles of parallel lines cut by a transversal.
( m angle 1 + m angle 2 + m angle 3 = 180^{circ} ) because they form a straight angle.
( m angle 5 + m angle 2 + m angle 6 = 180^{circ} ) by the substitution property of equality.
(c) ( m angle 1 = m angle 2 = m angle 3 = 60^{circ} )
( m angle 4 + m angle 5 = 180^{circ} ) and ( m angle 6 + m angle 7 = 180^{circ} )
( m angle 5 = m angle 6 = 60^{circ} )
( m angle 5 + m angle 2 + m angle 6 = 180^{circ} )

Explanation:

• For option A: $\angle4\cong\angle2$ is incorrect as $\angle4$ and $\angle2$ are not alternate - interior angles. $\angle4$ and $\angle1$ are alternate - interior angles (if we consider the left - hand transversal). Also, $\angle7\cong\angle2$ is wrong. $\angle7$ and $\angle3$ are alternate - interior angles (if we consider the right - hand transversal).
• For option B:
• Since line \(p\parallel q\), by the alternate - interior angles theorem (when a transversal intersects two parallel lines), \(\angle5\cong\angle1\) (left - hand transversal) and \(\angle6\cong\angle3\) (right - hand transversal).
• We know that \(m\angle1 + m\angle2+m\angle3 = 180^{\circ}\) because \(\angle1,\angle2,\angle3\) form a straight angle (\(180^{\circ}\)).
• By the substitution property of equality (substituting \(\angle1\) with \(\angle5\) and \(\angle3\) with \(\angle6\)), we get \(m\angle5 + m\angle2+m\angle6 = 180^{\circ}\), where \(\angle2,\angle5,\angle6\) are the interior angles of the triangle.
• For option C: We cannot assume that \(m\angle1 = m\angle2 = m\angle3=60^{\circ}\) as the triangle is not given to be equilateral. Also, \(m\angle4 + m\angle5 = 180^{\circ}\) and \(m\angle6 + m\angle7 = 180^{\circ}\) are linear - pair relationships, but this approach does not correctly use the parallel - line and angle - relationship properties to prove the sum of interior angles of a triangle.

Answer:

B. \(\angle5\cong\angle1\) and \(\angle6\cong\angle3\) because they are alternate interior angles of parallel lines cut by a transversal. \(m\angle1 + m\angle2 + m\angle3 = 180^{\circ}\) because they form a straight angle. \(m\angle5 + m\angle2 + m\angle6 = 180^{\circ}\) by the substitution property of equality.