How Are Angles Between Lines Related to Their Perpendiculars?

[Aladin]

Please explain the statement according to the diagram below.
"The angle between two lines is equal to the angles between their perpandiculars"
<i` = <r`
and
<i = <r
why ? How ?

 

[HallsofIvy]

Are you sure it is <i= <r' and <r= <i'?? Unless there is some information you are not giving, that's not true. i and r could be any angles at all but it is easy to show that <i= <i' and <r= <r'.

That's relatively straight forward geometry.

Angle i' in your picture is the angle of between the vertical axis and line A. It is, of course, the complement of the angle between line A and the horizontal axis (call that angle j) since together they form a right angle. But that is an angle in a right triangle in the right triangle formed by A, the line perpendicular to A, and the horizontal line. Since i is the other angle in the triangle, it is the complement of j. That is, i and i' are both complements to j and so are congruent.
Same argument for <r= <r'.

It's easy to see that the angle between the two lines is i'+ r' but the angle between their pependiculars is NOT i+ r: it is the supplement (180 degrees- (i+r). Your statement "The angle between two lines is equal to the angles between their perpendiculars" is not true. The angle between two lines is the supplement of the angle between two the two pependiculars. Geometrically, the two lines and their perpendiculars form a quadrilateral in which the "angle between the two lines" and the "sum of the angle between their perpendiculars" are opposite angles. Those angles are the same if and only if the quadrilateral is a parallelogram. In this case, since the angle between a line and its perpendicular is 90 degrees, the quadrilateral must be a rectangle.

The statement "The angle between two lines is equal to the angle between their perpendiculars" is true if and only if the angle between the two lines is a right angle.