Question

A) Two non-vertical lines have the same slope if and only if they are parallel.

B) If two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent.

C) Angles supplementary to the same angle or to congruent angles are congruent.

D) If two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent.

Answer 1

Answer: D) If two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent.

Explanation:

Since, By the Alternative interior angle theorem,

When two parallel lines are cut by the same transversal then the alternative interior angles made by the transversal on these parallel lines are congruent to each other.

Here AC ║ A'C'

⇒ AD ║ A'C'

⇒ parallel lines AD and A'C are cut by the same transversal A'D

⇒ ∠ CDB' ≅ ∠ BA'C' -------(1)

Similarly, When, AB ║ A'B

⇒ AB ║ A'D and AB ║EB'

⇒ Parallel lines AB and A'D are cut by the same transversal AD and Parallel lines AB and EB' are cu by the same transversal EB.

⇒ ∠ BAC ≅ ∠CDB' ------(2) and ∠ABC ≅ ∠BEB' -------(3)

Also, BC ║ B'C'

⇒ BE ║ B'C'

And, these paralle lines are cut by the same transversal B'E,

Thus, ∠BEB' ≅ ∠ A'B'C' ---------(4)

Thus, By equations (1) and (2),

∠ BAC ≅ ∠B'A'C ( By transitive property of equality )

And, By equation (3) and (4),

∠ABC ≅ ∠ A'B'C' ( By transitive property of equality )