a. Labeled diagram based on Priya and Han's conversation:
B 40
G 40
C(40)
b. No, there is nothing special about 40 degrees.
Theorem: If two lines are cut by a transversal such that the alternate interior angles are congruent, then the lines are parallel.
Proof:
Let lines l and m be cut by transversal t at points A and B, respectively.
Let ∠CAB≅∠ABM.
We want to prove that l∥m.
Draw line BX perpendicular to l at point B.
Since ∠ABX=90 ∘ , we have ∠CBX=180 ∘ −∠ABX−∠ABC=180 ∘ −90 ∘ −40 ∘ =50 ∘ .
Since ∠CBM=40 ∘ and ∠CBX=50 ∘ , we have ∠MBX=∠CBX−∠CBM=50 ∘ −40 ∘ =10 ∘ .
Since line BX is perpendicular to line l, we have ∠MBX+∠ABM=90 ∘ .
Substituting ∠MBX=10 ∘ into the equation above, we get 10 ∘ +∠ABM=90∘ .
Solving for ∠ABM, we get ∠ABM=80 ∘ .
Since ∠CAB≅∠ABM, we have ∠CAB=80 ∘ .
Since ∠CAB and ∠ABM are alternate interior angles, we have l∥m.
Therefore, if two lines are cut by a transversal such that the alternate interior angles are congruent, then the lines are parallel.