MHB Equality of Angles: 3 Equal Angles in a Picture

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The discussion centers on the equality of three angles in a geometric configuration, demonstrating that they are equal due to the properties of triangles and vertical angles. It explains that the sum of angles in a triangle is 180 degrees, leading to the conclusion that angles θ2 and θ3 are equal based on their relationships with vertical angles and right angles. The theorem stating that angles with mutually perpendicular sides are equal is highlighted as a key principle in this explanation. Additionally, the concept is noted for its practical applications in physics, particularly in problems involving inclined planes. Understanding these geometric relationships is essential for solving related mathematical and physical problems.
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Can you explain why the 3 angles in the picture are the same.
 

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There is a theorem saying that two angles whose sides are mutually perpendicular are equal.
 
Drain Brain said:
Can you explain why the 3 angles in the picture are the same.
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The sum of the angles of a triangle is equal to $180$.

The angles $\hat{F_1}$ and $\hat{F_2}$ are a pair of vertical angles, so they are equal, $\hat{F_1}=\hat{F_2}$

At the triangle $BFD$, the sum of the angles is:
$$\theta_2+\hat{F_2}+90 ^{\circ}=180^{\circ} \Rightarrow \theta_2+\hat{F_2}=90^{\circ} \ \ \ (1)$$

At the triangle $CEF$, the sum of the angles is:
$$\theta_3+\hat{F_1}+90^{\circ}=180 \Rightarrow \theta_3+\hat{F_1}=90^{\circ} \ \ \ (2)$$

$$\xrightarrow[(1)]{(2)} \theta_2+\hat{F_2}=\theta_3+\hat{F_1} \Rightarrow \theta_2=\theta_3$$

Then we do the same for the triangles $OFC$ and $BDF$ and we conclude that $\theta_1=\theta_2=\theta_3$
 

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Evgeny.Makarov said:
There is a theorem saying that two angles whose sides are mutually perpendicular are equal.

what do you mean? Can you show me a picture of mutually perpendicular angles? please bear with me. :) thanks!
 
I mean the following situation.

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If you consider angles composed of two rays (semi-lines) rather than lines infinite in both directions, then the angles may add up to $180^\circ$. But if you have two pairs of lines: $l_1,l_2$ and $l_1',l_2'$ such that $l_1\perp l_1'$ and $l_2\perp l_2'$ and if you consider the smaller angles formed by these lines, then these angles are equal.

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This property is especially useful in physics for solving problems with inclined plane.
 

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