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.

 

[tacman]

The statement "The angle between two lines is equal to the angles between their perpendiculars" is a geometric property that can be proven using the given diagram. In the diagram, we have two intersecting lines, <i and <r, which form four angles at the point of intersection. These angles are labeled as <i, <r, <i`, and <r`, with <i and <r being opposite angles and <i` and <r` being opposite angles.

Now, let's draw perpendicular lines from the point of intersection to each of the two lines. These perpendicular lines will form right angles with the original lines, and we can label them as <i_1 and <r_1. Since <i_1 and <r_1 are right angles, their measures are 90 degrees each.

Using basic geometry principles, we know that the sum of angles in a triangle is 180 degrees. Therefore, in triangle <i`<i_1<i, the sum of the angles must be 180 degrees. This can be written as <i` + <i_1 + <i = 180. Similarly, in triangle <r`<r_1<r, the sum of the angles must also be 180 degrees, which can be written as <r` + <r_1 + <r = 180.

Now, if we subtract <i` and <r` from both equations, we get <i_1 + <i = <r_1 + <r. But we know that <i_1 and <r_1 are both 90 degrees, so we can substitute them in the equation, giving us <i + <r = <i` + <r`. This means that the angles <i and <r are equal to the angles <i` and <r`, respectively.

Therefore, we can conclude that the angle between two lines, <i and <r, is equal to the angle between their perpendiculars, <i_1 and <r_1. This is why the statement is true and how it can be proven using the given diagram.