Question

5 transversal t cuts parallel lines a and b as shown in the diagram. which equation is necessarily true? o a. m∠1 = m∠2 o b. m∠4 = m∠6 o c. m∠1 + m∠5 = 90° o d. m∠3 + m∠5 = 180° o e. m∠7 = m∠8

Explanation:

Step1: Recall the property of parallel lines and transversal

When two parallel lines \(a\) and \(b\) are cut by a transversal \(t\), consecutive - interior angles (also known as same - side interior angles) are supplementary.

Step2: Identify consecutive - interior angles

\(\angle3\) and \(\angle5\) are consecutive - interior angles.

Step3: Apply the consecutive - interior angles theorem

By the consecutive - interior angles theorem, if \(a\parallel b\), then \(m\angle3 + m\angle5 = 180^{\circ}\)

For option A: \(\angle1\) and \(\angle2\) are adjacent angles, not necessarily equal.
For option B: \(\angle4\) and \(\angle6\) are not corresponding angles (corresponding angles are equal when lines are parallel, but \(\angle4\) and \(\angle6\) do not have a corresponding - angle relationship).
For option C: \(\angle1\) and \(\angle5\) are not complementary.
For option E: \(\angle7\) and \(\angle8\) are adjacent angles, not necessarily equal.

Answer:

D. \(m\angle3 + m\angle5=180^{\circ}\)