Question

1. complete the proof of the alternate exterior angles theorem.

given: ( ell parallel m ), cut by transversal ( t )
prove: ( angle 2 cong angle 7 )

statements

1. ( ell parallel m ), cut by transversal ( t )

2.

1. ( angle 6 cong angle 7 )

4.

reasons
1.

1. corresponding angles postulate

3.

1. transitive property of ( cong )

a. ( angle 2 cong angle 7 )
b. ( angle 2 cong angle 6 )
c. alternate interior angles theorem
d. vertical angles theorem
e. ( angle 2 cong angle 3 )
f. given

Explanation:

Step1: Fill in the reason for statement 1

The first statement is the given information. So the reason for statement 1 is "Given".

Step2: Use the Corresponding Angles Postulate

When two parallel lines \(l\parallel m\) are cut by a transversal \(t\), corresponding angles are congruent. \(\angle2\) and \(\angle6\) are corresponding angles. So statement 2 is \(\angle2\cong\angle6\).

Step3: Fill in the reason for statement 3

\(\angle6\) and \(\angle7\) are vertical angles. By the Vertical Angles Theorem, \(\angle6\cong\angle7\).

Step4: Use the Transitive Property

Since \(\angle2\cong\angle6\) (from step 2) and \(\angle6\cong\angle7\) (from step 3), by the Transitive Property of \(\cong\) (if \(a = b\) and \(b = c\), then \(a=c\)), we have \(\angle2\cong\angle7\).

Answer:

1. f. Given
2. b. \(\angle2\cong\angle6\)
3. d. Vertical Angles Theorem
4. a. \(\angle2\cong\angle7\)