Question

use the diagram to answer the question. line p is parallel to line q. which proof explains why the sum of the interior angles of the triangle must be 180°? a ∠4 ≅ ∠2 and ∠7 ≅ ∠2 because they are alternate interior angles of parallel lines cut by a transversal. m∠1 + m∠2 + m∠3 = 180° because they form a straight angle. m∠4 + m∠2 + m∠7 = 180° by the substitution property of equality. m∠5 + m∠2 + m∠6 = 180° by supplementary angles. b ∠5 ≅ ∠1 and ∠6 ≅ ∠3 because they are alternate interior angles of parallel lines cut by a transversal. m∠1 + m∠2 + m∠3 = 180° because they form a straight angle. m∠5 + m∠2 + m∠6 = 180° by the substitution property of equality. c m∠1 = m∠2 = m∠3 = 60° m∠4 + m∠5 = 180° and m∠6 + m∠7 = 180° m∠5 = m∠6 = 60° m∠5 + m∠2 + m∠6 = 180°

Explanation:

Step1: Recall angle - parallel line relationships

When two parallel lines \(p\) and \(q\) are cut by transversals, alternate - interior angles are equal. \(\angle5\) and \(\angle1\) are alternate - interior angles, and \(\angle6\) and \(\angle3\) are alternate - interior angles.

Step2: Use straight - angle property

\(\angle1+\angle2+\angle3 = 180^{\circ}\) because they form a straight angle.

Step3: Apply substitution

Since \(\angle5\cong\angle1\) and \(\angle6\cong\angle3\) (alternate - interior angles of parallel lines), by the substitution property of equality, we can replace \(\angle1\) with \(\angle5\) and \(\angle3\) with \(\angle6\) in the equation \(\angle1+\angle2+\angle3 = 180^{\circ}\), getting \(\angle5+\angle2+\angle6 = 180^{\circ}\), where \(\angle5\), \(\angle2\), and \(\angle6\) are the interior angles of the 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.