Answer:
#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
