Question

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

Explanation:

Step1: Recall properties of parallel lines and transversal

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

Step2: Analyze each option

• Option A: \(m\angle1 + m\angle2=180^{\circ}\) (linear pair), so \(m\angle1

eq m\angle2\) unless \(t\) is perpendicular to \(a\).

• Option B: \(\angle4\) and \(\angle6\) are alternate - interior angles. Since \(a\parallel b\) and \(t\) is a transversal, by the alternate - interior angles theorem, \(m\angle4 = m\angle6\).
• Option C: There is no property that directly gives \(m\angle1 + m\angle5 = 90^{\circ}\).
• Option D: \(\angle3\) and \(\angle5\) are not same - side interior angles. \(\angle3\) and \(\angle5\) are not supplementary.
• Option E: \(m\angle7 + m\angle8=180^{\circ}\) (linear pair), so \(m\angle7

eq m\angle8\) unless \(t\) is perpendicular to \(b\).

Answer:

B. \(m\angle4 = m\angle6\)