Question

given: t intersects l and m ; t ⊥ l ; l || m. prove: m∠5 = m∠6 statements reasons 1. t intersects l and m ; t ⊥ l ; l || m 1. given 2. t ⊥ m 2. 3. m∠5 = 90° ; m∠6 = 90° 3. def. ⊥ ∠s 4. 4. substitution a. m∠5 = m∠6 b. perpendicular transversal theorem c. m∠5 = m∠2 d. theorem 3.4a

Explanation:

Step1: Determine the reason for \(t\perp m\)

Since \(t\perp l\) and \(l\parallel m\), by the Perpendicular Transversal Theorem (if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other line), we have \(t\perp m\). So the reason for statement 2 is "Perpendicular Transversal Theorem".

Step2: Determine the statement for substitution

We know \(m\angle5 = 90^{\circ}\) and \(m\angle6=90^{\circ}\). By substitution (if \(a = c\) and \(b = c\), then \(a = b\)), we substitute \(c = 90^{\circ}\), \(a=m\angle5\) and \(b = m\angle6\). So the statement for step 4 is \(m\angle5=m\angle6\).

Answer:

1. b; 4. a