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.

Explanation:

• En la opción A, hay un error en la identificación de ángulos alternos internos.
• En la opción B, se identifican correctamente los ángulos alternos internos (\(\angle5\cong\angle1\) y \(\angle6\cong\angle3\)) debido a que \(p\parallel q\) y se cortan por transversales.
• Sabemos que \(m\angle1 + m\angle2 + m\angle3=180^{\circ}\) porque forman un ángulo recto.
• Sustituyendo \(\angle1\) por \(\angle5\) y \(\angle3\) por \(\angle6\) (por la propiedad de sustitución de la igualdad) obtenemos \(m\angle5 + m\angle2 + m\angle6 = 180^{\circ}\), lo cual son los ángulos internos del triángulo.

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.