Name and Details of relationship of angles

Question

Could anyone tell me what this relationship of angles is called and where I can read more about it? I'm not mathematically strong but simply put, if a line is drawn perpendicular to the hypotenuse of a right angled triangle, that angle between that and a vertical line is equal to the angle opposite along the adjacent side to the right angle.

Thanks in advance

Answer 1

I don't think it has a name, but the form I heard is this:

• If two angles have perpendicular sides, they are either equal or 180° complements.
• If two angles have parallel sides, they are either equal or 180° complements.

I heard it in the context of a physics class since the property is useful for finding angles involving inclined planes, etc.

Note that the angles might be complements if you only know the sides are perpendicular. I found this book: http://www.chegg.com/homework-help/two-angles-respective-sides-perpendicular-angles-either-cong-chapter-4.1-problem-22e-solution-9780321830951-exc

If you would just like to focus on the scenario above, here's a brief proof:

∠ABC = ∠A + 90° (exterior angle property)

∠B = ∠ABC - 90°

∠A = ∠B

Answer 2

Note: The same is true independent if the big triangle has a $90^\circ$ angle or not, or even if it is a triangle. All you need is the tilted line and the horizontal.

Let's have the points $A$ and $B$ where you have the labels for the angles Extend the perpendicular to the tilted line until it intersects the horizontal. Call this $C$. Also, draw the vertical line from $B$. The intersection with $AC$ is $D$. Since angle $B$ and angle $\angle CBD$ are formed by the same lines and are opposite, you have $\angle B=\angle CBD$. The $CBD$ triangle has the angle at $D$ at $90^\circ$, so $\angle BCD=90^\circ-\angle B$. $ABC$ triangle is also a right angle triangle (at $B$), so $\angle A=90^\circ-\angle BCA=90^\circ-(90^\circ-\angle B)=\angle B$.