Question

in the diagram, transversal t cuts parallel lines a and b. which equation is necessarily true?
a. ( mangle1 = mangle7 )
b. ( mangle3 = mangle6 )
c. ( mangle5 + mangle8 = 90^{circ} )
d. ( mangle6 + mangle7 = 180^{circ} )

Explanation:

Step1: Recall properties of parallel lines and transversals

When a transversal cuts parallel lines, alternate - interior angles are equal.

Step2: Analyze each option

• Option A: \(m\angle1=m\angle7\) is not necessarily true. \(\angle1\) and \(\angle7\) are not in a standard equal - angle relationship (like alternate - interior, corresponding).
• Option B: Since \(a\parallel b\) and \(t\) is a transversal, \(\angle3\) and \(\angle6\) are alternate - interior angles. By the alternate - interior angles theorem, if two parallel lines are cut by a transversal, then alternate - interior angles are congruent. So \(m\angle3 = m\angle6\).
• Option C: \(\angle5\) and \(\angle8\) are vertical angles. \(m\angle5=m\angle8\), and there is no reason \(m\angle5 + m\angle8=90^{\circ}\) (they are not complementary unless specified).
• Option D: \(\angle6\) and \(\angle7\) are adjacent angles. \(m\angle6 + m\angle7 = 180^{\circ}\) would be true if they were supplementary (linear pair), but \(\angle6\) and \(\angle7\) are vertical angles (\(m\angle6=m\angle7\)) not supplementary (unless \(m\angle6 = m\angle7=90^{\circ}\), which is not given).

Answer:

B. \(m\angle3 = m\angle6\)