We can see that the two marked angles form a pair of consecutive interior angles. This means that we can use the converse of consecutive interior angles theorem to check whether or not we have parallel lines.

If we add the measures of the consecutive interior angles together we get:

124\degree+46\degree=170\degree

Since the sum of the measures is not 180\degree, the two angles are not supplementary.

The converse of consecutive interior angles states that the lines are parallel if the consecutive interior angles are supplementary. Since they are not supplementary, the lines are not parallel.

Therefore, there is not a pair of parallel lines in the figure.

We can see that the two marked angles form a pair of alternate exterior angles. For there to be a pair of parallel lines, the converse of alternate exterior angles theorem tells us that the two marked angles must be congruent.

We know that the two marked angles must be congruent, so we can set their measures to be equal and solve for x.

Therefore, the figure contains a pair of parallel lines when x=24.

For any other value of x, the equality would not be true and the lines would not be parallel.

No, we do not have enough information to conclude a \parallel c.

Given that \angle 1 \cong \angle 3, we could conclude that a \parallel b, using the converse of corresponding angles postulate. However, we do not have enough information to the conclude that a \parallel c. We would need some information involving \angle 4 to draw a conclusion.

There are multiple different ways to construct parallel lines using a compass and ruler, but all involve duplicating some object from the original line onto the new line.

One possible method that Adrian can use is the following:

Step 0: Choose a point A through which to construct a line parallel to \overleftrightarrow{BC}.

Step 1: Set the compass width to the distance AC.

Step 2: Construct an arc centered at B with the radius AC.

Step 3: Set the compass width to the distance AB.

Step 4: Construct an arc centered at C with the radius AB.

Step 5: Construct a line through A and the intersection of the two arcs. This line is parallel to \overleftrightarrow{BC}.

For this example, the duplicated object is the sides of the triangle \triangle ABC, such that the new constructed point will have the same distance from \overleftrightarrow{BC} as point A.

The result of these constructions is that we have two triangles \triangle ABC and \triangle A\rq CB which have all the same dimensions.

Since the base side \overline{BC} is the same for both triangles, their vertices A and A\rq will be at the same height, so then \overline{AA\rq} will be parallel to \overline{BC}, as required.