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9/15 II
A triangle has \( 3 \) sides and angles and is the smallest polygon.
The violet and pink angles have alternate and congruent angles at the oponent parallel line. Together with the third blue angle they will add up to \( 180^{\circ} \).
With the knowlegde of \( 2 \) angles the third angle can be determined.
Move the blue button to show the exercise.
10/15 II
A first application of the sum of the inner angles is to determine the sum of the inner angles of a polygon.
Every higher polygon can be filled up with triangles, like it is shown with the
\( 12 \)-gon.
Since each triangle contains a sum of \( 180^{\circ} \), the sum of all triangles is the total sum of the inner angles of the polygon.
For a uniform polygon, a single angle can be determined by dividing the total sum by the number of the corners.
11/15 II
Another application of the sum of the inner angles is an isosceles triangle, which has two sides of equal length.
An isosceles triangle is shown inside the semicircle in the figure. Two of its sides are given by the radius and stay of equal length when the triangle is moved inside the semicircle.
Since its two inner halfs are symmetric, the two angles have the same size.
12/15 II
A special case of a triangle occurs, when one of its angles becomes \( 90^{\circ} \). Then its called a right triangle.
Thale's theorem states, that a triangle in a semicircle always contains an angle of \( 90^{\circ} \).
The triangle in the figure consists of \( 2 \) isosceles triangles. The blue and violet angles add up to \( 180^{\circ} \) according to the sum of the inner angles.
A proof of thale's theorem is shown in the exercise.
sum of inner angles: \( 180^{\circ} \)
polygon, sum of angles: \((n-2)\cdot 180^{\circ}\)
isosceles triangle: two angles have the same size
thale's theorem: a triangle in a semicircle is a right triangle
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\( \alpha = \) \(^{\circ}\)
\( \beta = \) \(^{\circ}\)
Determine the third angle:
\( \gamma = \) \(^{\circ}\)check
Move the blue button again for the next exercise.
\( \alpha = \) \(^{\circ}\)
\( \beta = \) \(^{\circ}\)
Determine the third angle:
\( \gamma = \) \(^{\circ}\)check
\( \alpha = 48.9^{\circ}\)
\( \beta = 107.6^{\circ}\)
Determine the third angle:
\( \gamma = \) \(^{\circ}\)check
latex
48.9^{\circ}+107.6^{\circ}+23.5^{\circ}
=180^{\circ}
\( sum = \) \(^{\circ}\)check
Determine one inner angle of a uniform 12-gon:
\( 1\>angle = \) \(^{\circ}\)check
Determine the sum of all inner angles of a 24-gon:
\( sum = \) \(^{\circ}\)check
Determine one inner angle of a uniform 24-gon:
\( 1\>angle = \) \(^{\circ}\)check
latex
22\cdot180^{\circ}=3960^{\circ}
latex
\frac{22\cdot180^{\circ}}{24}=165^{\circ}
\( \gamma = \) \(^{\circ}\)
\( \alpha = \beta =\) \(^{\circ}\)check
Determine the angles for:
\( \gamma = 21.0^{\circ}\)
\( \alpha = \beta =\) \(^{\circ}\)check
latex
21.0^{\circ}+2\cdot79.5^{\circ}
=180^{\circ}
latex
\alpha+\beta+(\alpha+\beta)=180^{\circ}
the latex code for multiplication is: \cdot
Divide both sides by 2:
\( \alpha = \)
45.0
\(^{\circ}\)
\( \beta = \)
45.0
\(^{\circ}\)
end of example