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 Euclid's Parallel Postulate, lines \(m\) and \(n\) are not parallel.

Step1: Analyze the direction of lines

Non - parallel lines (intersecting lines) will intersect on the side where the sum of the interior angles is less than \(180^\circ\) (or we can think about the "closing" side of the angles). The sum of the angles \(64^\circ\) and \(114^\circ\) is \(178^\circ\), which is less than \(180^\circ\). The angle of \(64^\circ\) and \(114^\circ\) are on the left - hand side of the transversal (from the orientation of the diagram). So, lines \(m\) and \(n\) will intersect on the left side of the transversal.

Step1: Recall the parallel line criterion

We can use Euclid's Parallel Postulate. For two lines cut by a transversal, if the sum of the interior angles on the same side of the transversal is equal to \(180^\circ\) (two right angles), the lines are parallel; if the sum is not equal to \(180^\circ\), the lines are non - parallel and will intersect. So, to determine if two non - parallel lines will intersect, we can check the sum of the interior angles formed by a transversal with the two lines. If the sum of the interior angles on the same side of the transversal is not \(180^\circ\), the lines are non - parallel and will intersect. We can also use other angle - related criteria like corresponding angles, alternate interior angles. If corresponding angles are not equal, or alternate interior angles are not equal, the lines are non - parallel and will intersect.

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)