[Math] Without using angle measure how to prove two lines are parallel to the same line are parallel to each other

euclidean-geometrygeometry

I am currently using the book "Euclidean Geometry" by David M. Clark.

I have to prove that: two lines parallel to the same line are parallel to each other.

I am not allowed to use angle measure yet (degrees).

I am able to use any triangle congruence (SSS, SAS, AAS, ASA, HL)

I am allowed to use angle bisectors, midpoints, circles, right angles, isosceles triangles, vertical angles, corresponding angles, alternate interior angles, exterior angles, and squares to prove this.

We have these theorems which may be useful in proving this:

  • If two lines have a transversal which forms alternative interior
    angles that are congruent, then the two lines are parallel.

  • If two lines have a transversal which forms corresponding angles that
    are congruent, then the two lines are parallel.

  • An exterior angle of a transversal is not congruent to either
    opposite interior angle.

  • Vertical angles are congruent.

  • Congruent angles have congruent supplements.

  • All right angles are congruent.

  • Axiom 5: For every line l and every point P not on l, there is
    at most one line containing P that is parallel to l.

I started off drawing two parallel lines, l and m and point P on m, but I really don't even know where to begin.

Any help would be appreciated.

Best Answer

I think I have a correct proof now.

Proof by contradiction: Assume to the contrary that two lines parallel to the same line are not parallel to each other. Without loss of generality, assume line m and line n are parallel to a line l, but m and n are not parallel to each other. Then, m and n intersect at a point, P that is not on line l. However, this contradicts Axiom 5 because two lines would be containing P and be parallel to l. So the assumption that m and n are not parallel was incorrect. Thus, m and n are parallel to l and also parallel to each other.

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