1- Write the following proof. Given: a parallel to b, a perpendicular to c, and b perpendicular to d Prove: c parallel to d Statements                           Reasons

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embizze | High School Teacher | (Level 1) Educator Emeritus

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(1) `a || b `              (1) Given

(2) `a _|_ c `            (2) Given

(3) `b _|_c `            (3) If one of two parallel lines is perpendicular to a line, then the other line is perpendicular

(4) `b _|_ d `            (4) Given

(5) `c || d `              (5) In a plane, if two lines are perpendicular to the same line, then they are parallel

** Depending on your instructor, you may have to use the fact that you can assume that the diagram lies in one plane, in which case you would include a line like diagram as given.

** It is critical that you include, in the reason for step 5, that we are talking about two lines in a plane. Otherwise this statement is false. Imagine a vertical pole with "legs" jutting out in different directions at different heights. Each "leg" is perpendicular to the pole, but skew to the other "legs".

** You should be aware that, despite most secondary math texts, you will probably never see a two column proof outside of a secondary geometry course. Most proofs are written in paragraph form, as justaguide has done. Realize that a proof is just a convincing argument, where every statement can be shown to be true. Depending on your audience, you might leave out the reasons for each step, as a college text often does. You should compare the two-column proof to the paragraph proof and see if you could construct a two-column proof from the words.

justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

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In the problem, it is given that line a is parallel to line b, line a is perpendicular to line c and line b is perpendicular to line d.

Line a is perpendicular to line c, as a result the angle marked is 90 degrees. For the lines a and b with transversal line c, A' = 90 as corresponding angles are equal in the case of parallel lines.

Line b is perpendicular to line d, as a result A'' = 90 degrees.

For lines c and d with b as transversal, the corresponding angles A' and A'' are both equal to 90 degrees. If the corresponding angles for two lines are equal, the lines are parallel. This proves that line c is parallel to line d.

justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

In the problem it is given that line a is parallel to line b, line a is perpendicular to line c and line b is perpendicular to line d.

As line a is perpendicular to line c, the angle marked in the figure has a measure of 90 degrees. Line a is parallel to b, as corresponding angles are equal in the case of parallel lines, A' = 90 degrees. Line b and line d are perpendicular, as a result, the marked angle is 90 degrees. A' and 90 degrees are supplementary angles, as a result A'' = 180 - 90 = 90 degrees.

As A' and A'' are both equal to 90 degrees, for the lines c and d with b as transversal, the corresponding angles are equal. This indicates that lines c and d are parallel.

The equivalence of corresponding angles proves that lines c and d are parallel.

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