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APPENDIX - Proof of the Z Theorem

( \newcommand{\kernel}{\mathrm{null}\,}\)

In section 1.4 we stated but did not prove the following theorem:

Theorem 1 (The "Z" Theorem)

If two lines are parallel then their alternate interior angles are equal. If the alternate interior angles of two lines are equal then the lines must be parallel.

Theorem 1 consists of two statements, each one the converse of the other, We will prove the second statement first:

Theorem 1 (second part).

If the alternate interior angles of two lines are equal then the lines must be parallel. In Figure 1, if x=x then AB must be parallel to CD.

Screen Shot 2021-01-19 at 8.50.29 PM.png

Figure 1. We will prove that if x=x then AB is parallel to CD.

Proof

Suppose x=x and AB is not parallel to CD. This means that AB and CD meet at some point G, as in Figure 2, forming a triangle, KLG.

Screen Shot 2021-01-19 at 8.57.36 PM.png

Figure 2. If AB is not parallel to CD then they must meet at some point G.

We know from the discussion preceding the ASA Theorem (THEOREM 2.3.1, section 2.3) that KLG can be constructed from the two angles x and y and the included side KL. Now x=x (by assumption) and y=y (y=180x=180x=y). Therefore by the same construction x and y when extended should yield a triangle congruent to KLG. Call this new triangle LKH (see Figure 3). We now have that AB and CD are both straight lines through G and H. This is impossible since one and only one distinct straight line can be drawn through two points. Therefore our assumption that AB is not parallel to CD is incorrect, that is, AB must be parallel to CD. This completes the proof.

Screen Shot 2021-01-19 at 9.04.21 PM.png

Figure 3. Since x=x and y=y, x and y when extended should also form a triangle, LXH.

Theorem 1 (first part)

If two lines are parallel then their alternate interior angles are equal. In Figure 4, if AB is parallel to CD then x=x.

Screen Shot 2021-01-19 at 9.10.19 PM.png

Figure 4. We will prove that if AB is parallel to CD then x=x.

Proof

Suppose AB is parallel to CD and xx. One of the angles is larger; suppose it is x that is larger. Draw EF through P so that w=w as in Figure 5. EF is parallel to CD because we have just prove (THEOREM 1, second part) that two lines are parallel if their alternate interior angles are equal. This contradicts the parallel postulate (section 1.4) which states that through a point not on a given line (here point P and line CD) one and only one line can be drawn parallel to the given line. Therefore x must be equal to x. This completes the proof of THEOREM 1.

Screen Shot 2021-01-19 at 9.15.34 PM.png

Figure 5. Draw EF so that w=w.

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