Question

1. euclid’s parallel postulate (read it again from the class work) is sometimes phrased as the following: two straight lines that are crossed by a transversal will only be parallel if the sum of the two interior angles formed on the same side of the transversal is equivalent to the sum of two right angles. (a) in the diagram shown, lines m and n are crossed by transversal line t. why are m and n not parallel? (b) on which side, the right or left of the transversal will m and n intersect? (c) can you determine a way to know, based on (b), on which side of a transversal two non - parallel lines will intersect? diagram with lines m, n, t, angles 64° and 114°

Explanation:

Part (a)

Step1: Recall the parallel line condition

For two lines cut by a transversal, if the sum of the interior angles on the same side is \(180^\circ\) (two right angles, since \(90^\circ\times2 = 180^\circ\)), the lines are parallel. Here, we have angles \(64^\circ\) and \(114^\circ\).

Step2: Calculate the sum of the angles

Sum the two angles: \(64^\circ+ 114^\circ=178^\circ\).

Step3: Compare with \(180^\circ\)

Since \(178^\circ
eq180^\circ\), the sum of the interior angles on the same side of the transversal is not equal to two right angles. So, by the given postulate, lines \(m\) and \(n\) are not parallel.

Step1: Analyze the direction of non - parallel lines

Non - parallel lines (in a plane) will intersect. For two lines cut by a transversal, if they are not parallel, we can look at the angles. The angle of \(114^\circ\) and \(64^\circ\) are on one side (let's assume the left - right based on the diagram). Since the sum of angles on one side is less than \(180^\circ\), the lines will converge (intersect) on the side where the sum of the interior angles is less than \(180^\circ\). Let's check the sum on the other side. The supplementary angle of \(64^\circ\) is \(180 - 64=116^\circ\), and the supplementary angle of \(114^\circ\) is \(180 - 114 = 66^\circ\). The sum on the other side would be \(116+66 = 182^\circ\). Since the sum on the side with \(64^\circ\) and \(114^\circ\) is \(178^\circ<180^\circ\), the lines will intersect on the side of the transversal where these two angles (\(64^\circ\) and \(114^\circ\)) are located. From the diagram, if we consider the position of the angles, the lines \(m\) and \(n\) will intersect on the left side of the transversal \(t\) (or the side where the \(64^\circ\) and \(114^\circ\) angles are drawn, depending on the diagram's orientation).

Step1: Recall the parallel line postulate

Euclid's parallel postulate (in the given phrasing) states that two straight lines crossed by a transversal are parallel if and only if the sum of the two interior angles on the same side of the transversal is equal to two right angles (\(180^\circ\)).

Step2: Apply the postulate to non - parallel lines

For two non - parallel lines cut by a transversal, the sum of the interior angles on the same side of the transversal is not equal to \(180^\circ\). If the sum of the interior angles on one side of the transversal is less than \(180^\circ\), the lines will intersect on that side (because the angles "push" the lines towards each other on that side). If the sum is greater than \(180^\circ\), the lines will intersect on the opposite side (since the supplementary angles on the opposite side will have a sum less than \(180^\circ\)). So, by checking the sum of the interior angles on the same side of the transversal: if the sum is not \(180^\circ\), we can determine the side of intersection. If the sum \(S<180^\circ\), the lines intersect on the side where the angles with sum \(S\) are located; if \(S > 180^\circ\), they intersect on the opposite side (because the sum of the angles on the opposite side will be \(360 - S\), and if \(S>180\), then \(360 - S<180\)).

Answer:

Lines \(m\) and \(n\) are not parallel because the sum of the interior angles on the same side of transversal \(t\) (\(64^\circ + 114^\circ=178^\circ\)) is not equal to \(180^\circ\) (two right angles).

Part (b)