Answer:
See below.
Step-by-step explanation:
Vertical Angles Theorem
When two straight lines intersect, the opposite vertical angles are congruent.
Alternate Interior Angles Theorem
If a line intersects a set of parallel lines in the same plane at two distinct points, the alternate interior angles that are formed are congruent.
Transitive Property of Equality
If a=b and c=b, then a=c.
Proof that ∠1=∠2
- ∠3 is equal to ∠4 (Vertical Angle Theorem).
- As BC intersects the set of parallel lines AB and CD (given), ∠3 is equal to ∠2 (Alternate Interior Angles Theorem).
- If ∠3=∠4 and ∠3=∠2 then ∠2=∠4 (Transitive Property of Equality).
- Given that ∠1=∠4 and ∠2=∠4 then ∠1=∠2 (Transitive Property of Equality).
[tex]\begin{array}{c|c}\sf Statement & \sf Reason\\\cline{1-2}\\ \angle 3 = \angle 4 & \textsf{Vertical Angle Theorem}\\\\AB \parallel CD & \textsf{Given}\\\\\angle 3 = \angle 2 & \textsf{Alternate Interior Angles Theorem}\\\\\angle 2 = \angle 4 & \textsf{Transitive Property of Equality}\\\\\angle 1 = \angle 4 & \textsf{Given}\\\\\angle 1 = \angle 2 & \textsf{Transitive Property of Equality}\\\\\end{array}[/tex]
