Question

in the diagram, transversal (l) cuts parallel lines (a) and (b). which equation is necessarily true? a. (mangle1 = mangle7) b. (mangle3 = mangle6) c. (mangle5 + mangle8=180^{circ}) d. (mangle6 + mangle7 = 180^{circ})

Explanation:

Step1: Recall parallel - lines and transversal properties

When a transversal cuts parallel lines, vertical angles are equal, corresponding angles are equal, alternate - interior angles are equal, and same - side interior angles are supplementary.

Step2: Analyze option A

$\angle1$ and $\angle7$ are not corresponding, alternate - interior, or vertical angles. They have no fixed equal relationship.

Step3: Analyze option B

$\angle3$ and $\angle6$ are alternate - interior angles. When a transversal cuts parallel lines, alternate - interior angles are equal. So $m\angle3=m\angle6$.

Step4: Analyze option C

$\angle5$ and $\angle8$ are vertical angles. Vertical angles are equal, not complementary ($m\angle5 + m\angle8=180^{\circ}$ is wrong for vertical angles).

Step5: Analyze option D

$\angle6$ and $\angle7$ are vertical angles. Vertical angles are equal, not supplementary.

Answer:

B. $m\angle3 = m\angle6$