. In the given figure, are the lines a and b parallel? Justify your answer.
The Correct Answer and Explanation is:
Answer:
No, the lines a and b are not parallel.
Justification:
To determine if two lines are parallel when they are intersected by a third line, called a transversal, we can examine the relationships between the angles formed. One key principle is the Converse of the Consecutive Interior Angles Theorem. This theorem states that if the two interior angles that lie on the same side of the transversal are supplementary, meaning their sum is 180°, then the two lines are parallel.
Let’s analyze the angles given in the figure to apply this principle.
- We are given an interior angle of 125° at the intersection of the transversal and line b. This angle is on the left side of the transversal.
- We need to find the other interior angle on the same side of the transversal, which is located at the intersection with line a. The figure shows an exterior angle of 75°. The angle that is vertically opposite to this 75° angle is also 75°. This vertically opposite angle is an interior angle located on the left side of the transversal. Vertically opposite angles are always equal.
- Now we have the two consecutive interior angles: one is 125° and the other is 75°.
- According to the theorem, if lines a and b were parallel, the sum of these two angles should be 180°. Let’s calculate their sum:
75° + 125° = 200° - The sum of the consecutive interior angles is 200°, which is not equal to 180°.
Because the sum of the consecutive interior angles is not 180°, the condition for the lines to be parallel is not met. Therefore, we can conclude with certainty that line a is not parallel to line b.
