Question

Відношення двох внутрішніх кутів трикутника дорівнює 2:3, а зовнішніх кутів при цих же вершинах 11:9. Знайти в градусах третій внутрішній кут трикутника.

Answer 1

Чертеж во вложении.
Обозначим ∠В=х.
Т.к. ∠ВАС:∠ВСА=2:3, то пусть ∠ВАС=2t, ∠ВСА=3t.
Т.к. ∠МАВ:∠КСВ=11:9, то пусть ∠МАВ=11к, ∠КСВ=9к.
По свойству внешнего угла треугольника получим два равенства:
11k=х+3t
9k=x+2t
По теореме о сумме углов треугольника х=180°-(2t+3t)=180°-5t.
Решаем систему уравнений:
$\begin{cases} 11k=x+3t \\9k=x+2t \\ x=180-5t \end{cases} <=> \begin{cases} t=2k \\ x=180-10k \\ x=11k-3t=11k-6k=5k \end{cases} => \\ 5k=180-10k\\ 15k=180\\ k=12\\\\ \angle B=180^o-10*12^o=60^o$
Ответ: ∠В=60°.

Answer 2

Finding the Third Interior Angle of a Triangle

To find the third interior angle of a triangle, we can use the fact that the sum of the interior angles of any triangle is always 180 degrees.

Let's denote the two given interior angles as 2x and 3x. According to the problem statement, the ratio of these two angles is 2:3. Therefore, we can write the equation:

2x + 3x + Third Angle = 180 degrees

Simplifying the equation, we have:

5x + Third Angle = 180 degrees

Now, let's consider the relationship between the exterior angles and the interior angles of a triangle. The exterior angle at a vertex of a triangle is equal to the sum of the two non-adjacent interior angles at that vertex. In this case, the ratio of the exterior angles is given as 11:9.

Let's denote the two given exterior angles as 11y and 9y. According to the problem statement, the ratio of these two angles is 11:9. Therefore, we can write the equation:

11y + 9y + Third Exterior Angle = 360 degrees

Simplifying the equation, we have:

20y + Third Exterior Angle = 360 degrees

Now, let's solve these two equations simultaneously to find the values of x and y.

From the first equation, we have:

5x + Third Angle = 180 degrees

From the second equation, we have:

20y + Third Exterior Angle = 360 degrees

To solve these equations, we need more information or additional equations. The problem statement does not provide any further information or equations. Therefore, it is not possible to determine the values of x and y or the third interior angle of the triangle with the given information.

If you have any additional information or equations, please provide them, and I will be happy to help you further.