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Equality of Angles: 3 Equal Angles in a Picture
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In summary, the three angles in the picture are the same because of a theorem stating that two angles formed by mutually perpendicular lines are equal. This is shown by considering the sum of angles in two different triangles and using the fact that vertical angles are equal. This property is useful in solving problems involving inclined planes in physics.
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Evgeny.Makarov
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There is a theorem saying that two angles whose sides are mutually perpendicular are equal.
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mathmari
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View attachment 2891Drain Brain said:
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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bergausstein
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Evgeny.Makarov said:
what do you mean? Can you show me a picture of mutually perpendicular angles? please bear with me. :) thanks!
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Evgeny.Makarov
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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.
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.
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FAQ: Equality of Angles: 3 Equal Angles in a Picture
What does it mean for angles to be equal in a picture?
When angles are equal in a picture, it means that they have the same measure or degree. In other words, they have the same amount of rotation, and their sides are in the same position relative to each other.
How can you determine if three angles in a picture are equal?
To determine if three angles in a picture are equal, you can use a protractor to measure the angles. If all three angles have the same measurement, then they are equal. Alternatively, you can also compare the angles visually by looking at their shapes and positions.
What is the importance of equal angles in a picture?
Equal angles in a picture are important because they help us understand the relationships between different shapes and objects. They also allow us to make accurate measurements and calculations, which are essential in many fields such as mathematics, engineering, and architecture.
Can angles be equal even if they appear different in a picture?
Yes, angles can be equal even if they appear different in a picture. This is because the size and shape of an angle can vary depending on the scale or perspective of the picture. It is the measurement or degree of the angle that determines its equality, not its appearance.
What is the difference between equal angles and congruent angles?
Equal angles and congruent angles are often used interchangeably, but there is a subtle difference between the two. Equal angles have the same measurement, while congruent angles not only have the same measurement but also the same shape and size. In other words, congruent angles are identical to each other, while equal angles can be different in appearance.