Question

Биссектрисы прямого и острого углов прямоугольного треугольника при пересечении образуют углы, один из которых равен 54 градусам. найдите острые углы треугольника.

Answer 1

Построим прямоугольный треугольник ABC (С=90, угол А - острый). При пересечении двух биссектрис образуются смежные и вертикальные углы и назовем точку пересечения буквой К, следовательно два одинаковых и два разных угла. Пусть один из них будет 54 градуса (по условию), то второй угол равен 126 градусам. Так как биссектриса делит угол по полом, то половина прямого угла будет равна 45 градусам. Рассмотрим треугольник АСК. Угол С=45, угол К=126 => угол А=9градусам.
Рассмотрим треугольник АВС, угол А=18 градусам, В=72градусов. 

Answer 2

Finding the Angles of a Right-Angled Triangle

When the bisectors of the acute angles of a right-angled triangle intersect, they form angles, one of which is equal to 54 degrees. To find the acute angles of the triangle, we can use the properties of right-angled triangles and the information provided.

Given: - One of the angles formed by the bisectors is 54 degrees.

We know that in a right-angled triangle: - The sum of all angles is 180 degrees. - One angle is always 90 degrees.

Using this information, we can calculate the acute angles of the triangle.

Calculating the Acute Angles

Let's denote the acute angles of the right-angled triangle as A and B.

We know that the sum of all angles in a triangle is 180 degrees. Therefore, we can express the sum of the angles in the following equation: A + B + 90 = 180

Given that one of the angles formed by the bisectors is 54 degrees, we can express this information as: A/2 = 54

We can use these equations to solve for the values of A and B.

Solving for the Acute Angles

First, we can solve for A using the equation A/2 = 54: A = 2 * 54 A = 108

Now that we have the value of A, we can substitute it into the equation A + B + 90 = 180 to solve for B: 108 + B + 90 = 180 B = 180 - 108 - 90 B = 72

So, the acute angles of the right-angled triangle are: A = 108 degrees B = 72 degrees

Therefore, the acute angles of the right-angled triangle are 108 degrees and 72 degrees.

This information is based on the properties of right-angled triangles and the given angle formed by the bisectors