Question

use the figure below to answer the question. lines a and b are parallel. line c is a transversal. which statement is true? a. m∠1 ≠ m∠4, because they are opposite angles b. m∠1 = m∠6, because they are supplementary angles c. m∠1 ≠ m∠8, because they are adjacent angles d. m∠1 = m∠4, because they are vertical angles

Explanation:

Step1: Recall angle - pair relationships

When two lines intersect, vertical angles are equal. $\angle1$ and $\angle4$ are vertical angles formed by the intersection of line $a$ and line $c$.

Step2: Analyze each option

• Option A: $\angle1$ and $\angle4$ are vertical angles, not opposite angles in the non - mathematical sense used here, and $m\angle1 = m\angle4$, so A is false.
• Option B: $\angle1$ and $\angle6$ are neither supplementary (their sum is not necessarily 180 degrees) nor equal in general, so B is false.
• Option C: $\angle1$ and $\angle8$ are not adjacent angles. Adjacent angles share a common side and a common vertex. Also, in the context of parallel lines and transversals, they are alternate exterior angles and $m\angle1=m\angle8$ when $a\parallel b$, so C is false.
• Option D: $\angle1$ and $\angle4$ are vertical angles, and vertical angles are always equal, so $m\angle1 = m\angle4$.

Answer:

D. $m\angle1 = m\angle4$, because they are vertical angles