Question

a student wants to prove that the sum of the angles of triangle abc is 180°. he draws a line, xy, passing through vertex a and parallel to the side bc, as shown. which properties should he use for his proof?
○ m∠xab + m∠yac = 90°, and the measure of alternate interior angles are equal
○ m∠xab + m∠bac + m∠yac = 180°, and the measure of alternate interior angles are equal
○ m∠xab + m∠yac = 90°, and the measure of corresponding angles are equal
○ m∠xab + m∠bac + m∠yac = 180°, and the measure of corresponding angles are equal

Explanation:

Step1: Recall angle - line relationship

Since $XY$ is a straight - line, $\angle XAB+\angle BAC+\angle YAC = 180^{\circ}$ (by the definition of a straight - angle, the sum of angles on a straight line is $180^{\circ}$).

Step2: Use parallel - line properties

Because $XY\parallel BC$, we use the property of alternate interior angles. Alternate interior angles formed by a transversal ($AB$ and $AC$ are transversals for $XY$ and $BC$) and parallel lines are equal. This helps in relating the angles of the triangle to $\angle XAB$, $\angle BAC$, and $\angle YAC$.

Answer:

$m\angle XAB + m\angle BAC+m\angle YAC = 180^{\circ}$, and the measure of alternate interior angles are equal