Question

two parallel lines are crossed by a transversal. if m∠6 = 123.5°, then m∠1 is 56.5°. 67.5°. 123.5°. 136.5°. (diagram of two parallel lines and a transversal with angles labeled 1–8, lines w, x, y)

Explanation:

Step1: Identify Angle Relationships

When two parallel lines are cut by a transversal, corresponding angles are equal, and consecutive interior angles are supplementary. Also, vertical angles are equal, and linear pairs are supplementary. Here, ∠6 and ∠3 are same - side interior angles (supplementary), and ∠1 and ∠3 are vertical angles? Wait, no. Wait, ∠6 and ∠3: since lines x and y are parallel, and the transversal is the line with angles 1,2,3,4. Wait, actually, ∠6 and ∠3: let's see the diagram. ∠6 and ∠3 are same - side interior angles? Wait, no, ∠6 and ∠5 are a linear pair, ∠5 and ∠1: since x || y, ∠1 and ∠5 are corresponding angles? Wait, maybe a better approach: ∠6 and ∠3 are same - side interior angles (supplementary), so m∠3 + m∠6 = 180°. Then ∠1 and ∠3 are vertical angles? No, ∠1 and ∠3 are adjacent? Wait, no, ∠1 and ∠3: looking at the diagram, ∠1 and ∠3 are vertical angles? Wait, no, the angles around the intersection of line w and the other transversal: ∠1 and ∠3 are vertical angles? Wait, no, ∠1 and ∠3 are adjacent and form a linear pair? Wait, no, let's re - examine.

Wait, the two parallel lines are x and y, cut by transversal w (the line with angles 1,2,3,4) and another transversal (with angles 5,6,7,8). So ∠6 and ∠3: since x || y, ∠3 and ∠6 are same - side interior angles, so they are supplementary. So m∠3 = 180°−m∠6. Then ∠1 and ∠3: are they vertical angles? Wait, no, ∠1 and ∠3 are adjacent? Wait, no, ∠1 and ∠3: at the intersection of line w and the left - most parallel line (x), ∠1 and ∠3 are vertical angles? Wait, no, ∠1 and ∠3 are actually vertical angles? Wait, no, ∠1 and ∠3: if we have two intersecting lines, the vertical angles are equal. Wait, the line w intersects the parallel line x, forming angles 1,2,3,4. So ∠1 and ∠3 are vertical angles? No, ∠1 and ∠3 are adjacent and form a linear pair? Wait, no, ∠1 and ∠2 are a linear pair, ∠2 and ∠4 are vertical, ∠4 and ∠3 are a linear pair, ∠3 and ∠1 are vertical? Wait, maybe I made a mistake. Let's start over.

Given two parallel lines (x and y) cut by a transversal (the line with angles 1,2,3,4) and another transversal (with angles 5,6,7,8). ∠6 and ∠3: since x || y, ∠3 and ∠6 are same - side interior angles, so m∠3 + m∠6 = 180°. Then ∠1 and ∠3: are they corresponding? Wait, no, ∠1 and ∠5: since x || y, ∠1 and ∠5 are corresponding angles, so m∠1 = m∠5. And ∠5 and ∠6 are a linear pair, so m∠5 + m∠6 = 180°. Therefore, m∠1 + m∠6 = 180°.

Step2: Calculate m∠1

We know that m∠6 = 123.5°, and from the relationship m∠1 + m∠6 = 180°, we can solve for m∠1.

m∠1 = 180°−m∠6

Substitute m∠6 = 123.5° into the formula:

m∠1 = 180°−123.5° = 56.5°

Answer:

56.5°