Geometry Proofs with LinesDate: 2/6/96 at 15:46:33 From: Anonymous Subject: Geometric Proofs Prove THEOREM 3.3: If two lines are cut by a transversal so that two interior angles on the same side of the transversal are supplementary, then the lines are parallel. Prove: If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel. Date: 2/8/96 at 15:10:5 From: Doctor Ken Subject: Re: Geometric Proofs Hello! For these questions, I'll assume that you're working in good old Euclidean Geometry, as in normal high school Plane Geometry and not Hyperbolic or Spherical Geometry. For your first proof, try a proof by contradiction: assume that the two lines _aren't_ parallel (that they meet at some point P), and then show how that leads to a nonsense result. Hint: let the two points where the transversal intersects the two lines be called A and B. Then what is the sum of the angles in triangle ABP? For the second one, you can actually moosh it into another form of the first proof: / /x ------------------------/------------------------------- y/ z / ---------------------/---------------------------------- x / / We know that the two angles labeled "x" are congruent. Well, by vertical angles, we know that they have to be congruent to y, right? And the bottom x is supplementary to z, right? See if you can show that that means that y and z are supplementary. Good luck! -Doctor Ken, The Math Forum |
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