Question

given: lines ( ell ) and ( m ) cut by transversal ( t ), alternate exterior angles ( angle 1 cong angle 8 ) prove: ( ell parallel m ) statements reasons 1. 1. 2. ( angle 5 cong angle 8 ) 2. 3. 3. 4. ( ell parallel m ) 4. converse of the corresponding angles postulate a. lines ( ell ) and ( m ) cut by transversal ( t ), alternate exterior angles ( angle 1 cong angle 8 ) b. ( angle 1 cong angle 5 ) c. transitive property of congruence d. alternate interior angles theorem e. ( m angle 5 + m angle 8 = 180^{circ} ) f. given g. vertical angles theorem

Explanation:

Step1: Fill in the first statement and reason

The first statement should be the given information. So, statement 1 is "lines \(l\) and \(m\) cut by transversal \(t\), alternate exterior angles \(\angle1\cong\angle8\)" and the reason is "Given".

Step2: Explain \(\angle5\cong\angle8\)

\(\angle5\) and \(\angle8\) are vertical angles. By the Vertical Angles Theorem, \(\angle5\cong\angle8\). So, reason 2 is "Vertical Angles Theorem".

Step3: Use the Transitive Property

Since \(\angle1\cong\angle8\) (given) and \(\angle5\cong\angle8\) (from step 2), by the Transitive Property of Congruence, \(\angle1\cong\angle5\). So, statement 3 is "\(\angle1\cong\angle5\)" and reason 3 is "Transitive Property of Congruence".

Answer:

1. a, f
2. g
3. b, c