Question

match the image to the statement and determine whether \\(a \parallel b\\). press the arrows to expand each image.

a. ... by the converse of the alternate interior angles theorem.
b. ... by the converse of the alternate exterior angles theorem.
c. ... because consecutive (same-side) interior angles are not supplementary.
d. ... because corresponding angles are not congruent.

Explanation:

Analyze the first image

The first image shows two lines cut by a transversal. The marked angles are alternate interior angles, and they are marked as congruent with single arc ticks.
Using the Converse Statements concept, if alternate interior angles are congruent, then the lines are parallel. This matches statement a: "... by the Converse of the Alternate Interior Angles Theorem."

Analyze the second image

The second image shows two lines cut by a transversal. The marked angles are alternate exterior angles, with measures \(105^\circ\) and \(105^\circ\). Since \(105^\circ = 105^\circ\), these alternate exterior angles are congruent.
This matches statement b: "... by the Converse of the Alternate Exterior Angles Theorem."

Analyze the third image

The third image shows two lines cut by a transversal. The marked angles are consecutive (same-side) interior angles, with measures \(85^\circ\) and \(90^\circ\).
Using the Consecutive Interior Angles Theorem, for lines to be parallel, these angles must be supplementary. Here, \(85^\circ + 90^\circ = 175^\circ
eq 180^\circ\).
Thus, the lines are not parallel. This matches statement c: "... because consecutive (same-side) interior angles are not supplementary."

Analyze the fourth image

The fourth image shows two lines cut by a transversal. The marked angles are corresponding angles, but they have different tick marks (one has a single arc, the other has a double arc), indicating they are not congruent.
Thus, the lines are not parallel. This matches statement d: "... because corresponding angles are not congruent."

Answer:

Question 1

First image (alternate interior angles with congruent tick marks) matches:

• a. ... by the Converse of the Alternate Interior Angles Theorem. (Correct answer)

Question 2

Second image (alternate exterior angles measuring \(105^\circ\) and \(105^\circ\)) matches:

• b. ... by the Converse of the Alternate Exterior Angles Theorem. (Correct answer)

Question 3

Third image (consecutive interior angles measuring \(85^\circ\) and \(90^\circ\)) matches:

• c. ... because consecutive (same-side) interior angles are not supplementary. (Correct answer)

Question 4

Fourth image (corresponding angles with different tick marks) matches:

• d. ... because corresponding angles are not congruent. (Correct answer)