The first image is use for questions 1-3.

1.given ∠8≅∠12, which lines, if any, must be parallel based on the given information? justify your conclusion.

a∥b, converse of the corresponding angles theorem

c∥d, converse of the alternate exterior angles theorem

a∥b, converse of the same-side interior angles theorem

not enough information to make a conclusion

2.given ∠3≅∠13, which lines, if any, must be parallel based on the given information? justify your conclusion.

a∥b, converse of the alternate interior angles theorem

c∥d, converse of the same-side interior angles theorem

c∥d, converse of the corresponding angles theorem

not enough information to make a conclusion

3.given ∠11≅∠13.

which lines, if any, must be parallel based on the given information? justify your conclusion.

c∥d, converse of the alternate exterior angles theorem

a∥b, converse of the corresponding angles theorem

c∥d, converse of the same-side interior angles theorem

not enough information to make a conclusion

4.which segments are parallel?

select each correct answer.

ef

gh

ab

cd
four segments intersected by two transversals. the four segments from top to bottom are segment a b, segment e f, segment g h, and segment c d. the two transversals are segments a d and b c that intersect at point i. points a, e, i, h, and d lie on transversal a d. points b, f, i, g, and c lie on transversal b c. angle a is 34 degrees. angle a e f is 138 degrees. angle g h d is 146 degrees and angle d is 34 degrees.

5.for what value of x is line a parallel to line b?

enter your answer in the box.

x=

#1)  a∥b, Converse of the Corresponding Angles Theorem ; #2) a∥b, Converse of the Alternate Interior Angles Theorem ; #3) c∥d, Converse of the Alternate Exterior Angles Theorem ; #4) GH, AB, and CD; #5) 30

Explanation:

#1)  ∠8 and ∠12 are in the same relative position; both are below line c and to the left of lines a and b, respectively.  Being in the same relative position makes them corresponding angles; the converse of the corresponding angles theorem states that if two corresponding angles are congruent, then the lines are parallel.  Thus a and b are parallel.

#2) ∠3 and ∠13 are inside lines a and b, and are on opposite sides of line d.  This makes them alternate interior angles.  The converse of the alternate interior angles theorem states that if two alternate interior angles are congruent, then the lines are parallel.  Thus a and b are parallel.

#3) ∠11 and ∠13 are outside lines c and d, and on opposite sides of line a.  This makes them alternate exterior angles.  The converse of the alternate exterior angles theorem states that if two alternate exterior angles are congruent, then the lines must be parallel.  Thus c and d are parallel.

#4) ∠EAB and ∠IEF are corresponding angles, as they are in the same relative position.  The measure of ∠IEF is 180-138 = 42°; since this is not the same as the measure of ∠EAB, AB and EF are not parallel.

∠EAB and ∠IDC are alternate interior angles; this is because they are inside lines AB and CD and on opposite sides of the transversal AD.  Since they are congruent, AB and CD are parallel.

∠IHG and ∠HDC are corresponding angles, since they are in the same relative position. The measure of ∠IHG is 180-46=34°; since this is congruent to the measure of ∠HDC, GH and CD are congruent.

Thus AB, GH and CD are congruent.

#5) In order for a and b to be parallel, corresponding angles must be congruent.  This gives us the equation

2x+20 = 80

Subtract 20 from each side:

2x+20-20 = 80-20

2x = 60

Divide both sides by 2:

2x/2 = 60/2

x = 30