Given a pair of lines in the same plane, a transversal is a third line that intersects each of the two original lines, but not in the same point. Click on the Euclidean General Situation checkbox to see this situation illustrated for Euclidean Geometry. [br][br]Note that we can let the intersections of the transversal with the two original lines be X and Y respectively. This makes the brown line the transversal. [br][br][b]Angles[/b]: There are 8 different proper angles constructed with vertices at X or Y. Notice the labeling in the diagram. [br]There are various pairs of angles in the diagram.[br][br][b]Linear Pairs[/b]: Notice that there are several linear pairs of angles which must be supplementary (angle measures sum to pi radians = 180 degrees). The linear pairs of angles are A1 & A2, A2 & A3, A3 & A4, A4 & A1, A5 & A6, A6 & A7, A7 & A8, A8 & A5.[br][br][b]Vertical Angle Pairs[/b]: There are also several vertical angle pairs. These are always congruent. These are A1 & A3, A2 & A4, A5 & A7, A4 & A6[br][br][b]Corresponding Angles[/b]: There are pairs of corresponding angles. (e.g. A1 & A5 are corresponding because they are up and to the right of X and Y respectively). These pairs are: A1 & A5, A2 & A6, A3 & A7, A4 & A8.[br][br][b]Interior Angle[/b]s: The following are interior angles: A3, A4, A5, A6.[br][br][b]Exterior Angles[/b]: The following are exterior angles: A1, A2, A7, A8.[br][b][br]Alternate Interior Angle Pairs[/b]: A3 & A5, A2 & A6[br][b][br]Alternate Exterior Angle Pairs[/b]: A1 & A7, A2 & A8[br][br][b]Paris of Interior Angles on the Same Side of the Transversal[/b]: A3 & A6, A4 & A5[br]

Again, notice that there are 8 proper angles constructed with vertex at either X or Y. However, since vertical angles are congruent, this produces at most 4 different angle measures. Congruent angles are represented by the same color in the app. [br][br]Can you manipulate the controls so that there are fewer angle measures?[br][br]You should discover that there could be 4, 3, 2, or 1 different angle measures. [br][br]The typical situation gives 4 different angle measures. There are 3 different angle measures when the transversal is perpendicular to exactly one of the two lines. There is only one angle measure when the [br]transversal is perpendicular to both of the other lines. The really interesting case is when there are exactly 2 different angle measures. In Euclidean Geometry this happens exactly when the two (non-transversal) lines are parallel. Click on the Euclidean Parallel Situation checkbox to see this version illustrated.[br][br]In Euclidean Geometry the pair of lines are parallel if and only if there are two angle measures here. All the odd numbered angles are congruent: A1, A3, A5, A7 and all of the even numbered angles are congruent A2, A4, A6, A8. Furthermore, any even numbered angle and any odd numbered angle are supplementary in this parallel lines case. [br][br]Notice that in Euclidean Geometry there is exactly one line through Y that is parallel to the line through A and X.

Given a transversal to a pair of lines in Euclidean Geometry.[br][br][b]Z Property[/b]: The lines are parallel if and only if there is a pair of congruent alternate interior angles.[br](Check the Z Property checkbox to see this illustrated in the Euclidean Parallel Lines Situation.)[br][br][b]F Property[/b]: The lines are parallel if and only if there is a pair of congruent corresponding angles.[br](Check the F Property checkbox to see this illustrated in the Euclidean Parallel Lines Situation.)[br][br][b]C Property[/b]: The lines are parallel if and only if there is a pair of supplementary interior angles on the same side of the transversal.[br](Check the C Property checkbox to see this illustrated in the Euclidean Parallel Lines Situation.)[br][br][b]Alternate Exterior Angles[/b]: The lines are parallel if and only if there is a pair of congruent alternate exterior angles.[br][br][b]Exterior Angles[/b]: The lines are parallel if and only if there is a pair of supplementary exterior angles on the same side of the transversal.[br][br]If the hypotheses of any of the parts above are true, then they all are.

Now let's examine this situation in Hyperbolic Geometry. Here we are using the Half-plane model for Hyperbolic Geometry, where lines are semicircles, and angles are measured as the angle between tangent rays. Check on the Hyperbolic Geometry General Situation and the Hyperbolic Half-plane Edge check boxes to see this situation illustrated. [br][br]Notice that the labeling is the same. Once again, there are 8 proper angles formed with vertices at X and Y, but there are at most 4 different angle measures. We have exactly the same results for vertical angles and linear pairs that we saw in Euclidean Geometry.[br][br]Do the other results hold here as well? Manipulate the controls to investigate.[br][br]Check the Hyperbolic Geometry Z Parallel checkbox. With this construction we see that there is a pair of congruent alternate interior angles. In this case notice that the lines are parallel, and we end up with only two angle measures. Again all the even labeled angles are congruent, all of the odd labeled angles are congruent, and any even labeled angle is supplementary to any odd labeled angle.[br][br]However, notice that in Hyperbolic Geometry there are infinitely many lines through Y that are parallel to the line through A and X. Only one of these lines has congruent alternate interior angles. So, in Hyperbolic Geometry the Transversal to Parallel Lines Theorem above is only true in one direction. [br][br]

It is not hard to prove that if the hypothesis of any part of the theorem is true, then the hypothesis of every other part is also true. The key is prove the Z Property.[br][br]A. Assume that there is a transversal to a pair of lines m and n with the labeling in the diagrams, such that a pair of alternate interior angles are congruent, i.e. A3 is congruent to A5. [br][br]Now suppose that the lines are not parallel. Then there must be a point P where n and m intersect. We have triangle XYP. Notice that either A3 or A5 is an exterior angle of this triangle and the other is a remote interior angle. By the Exterior Angle Inequality Theorem, the remote exterior angle must have a greater measure than the remote interior angle. However, this contradicts the hypothesis, so the lines m and n must be parallel. (This proof is valid in both Euclidean and Hyperbolic Geometries.)[br][br]B. Assume that there is a transversal to a pair of lines m and n with the labeling in the diagrams, such that m and n are parallel. By the Angle Measure Postulate we can construct a unique ray with vertex Y so that it forms an alternate interior angle congruent to A3. By Part A this line must be parallel to line m. By the Euclidean Parallel Postulate there is exactly one line through Y parallel to m so this new line is actually n. Thus, A3 is congruent to A5. (This proof is valid in Euclidean Geometry, but not in Hyperbolic Geometry).