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
(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.
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.
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.