Equality of Angles: 3 Equal Angles in a Picture

[Drain Brain]

Can you explain why the 3 angles in the picture are the same.

 

[Evgeny.Makarov]

There is a theorem saying that two angles whose sides are mutually perpendicular are equal.

 

[mathmari]

Can you explain why the 3 angles in the picture are the same.

View attachment 2891

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 $\theta_1=\theta_2=\theta_3$

 

[bergausstein]

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!

 

[Evgeny.Makarov]

I mean the following situation.

View attachment 2923

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.

View attachment 2922

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