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Can someone provide a trigonometry/geometry insight to deduce the angle of the plane is the same as the angle of the component of the weight?

enter image description here

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BTW: If you are simply having trouble remembering which of the angles in the small triangle is the same as $\alpha$, imagine that there is hinge at the joint and let $\alpha$ get small... – dmckee♦ Jul 17 '11 at 13:23

2 Answers

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Angles with their sides perpendicular are always equal. In the present example the arrow of $\vec{F}_1$ is perpendicular to the baseline, and the longest dotted line is perpendicular to the incline.

You can just imagine rotating one of the two triangles to put it on top of the other. Since the sides start off perpendicular, after a 90-degree rotation they will align, and hence show you that the angles are equal.

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We have this theorem in Geometry:

Angles with perpendicular lines are equal.

But why? We can proof it. Consider these angles:

enter image description here

We have:

EIH + IHE + HEI = 180
GIB + IBG + BGI = 180

thus

EIH + IHE + HEI = GIB + IBG + BGI

But EHI = 90 = IGB, because lines are perpendicular. Also HIE = GIB because:

HIE + HIG = 180 = GIB + HIG => GIB = HIE

Therefore, we can remove equal values from both sides to get:

**HEI = IBG**
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Your notation if confusing, I don't understand it. – Vicfred Jul 17 '11 at 16:27
IHE is the right angle marked near the point H, with sides IH and HE. The sum of the 3 interior angles of a plane triangle equals 180º. More clear ? – Helder Velez Jul 18 '11 at 14:13
Thanks @Vicfred. It's a theorem. My apologies. – Saeed Neamati Jul 18 '11 at 16:53

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