Question

Внутрішню точку О трикутника АВС з’єднали з вершинами трикутника. Виявилось, що AOB=90 OB=OC ACB=45 Знайти величину кута ОАВ.

Answer 1

Вітаю.

Відповідь: 45°

Answer 2

Problem Analysis

We are given a triangle ABC and a point O inside the triangle. The angles AOB, OB, and ACB are known to be 90 degrees, 45 degrees, and 45 degrees, respectively. We need to find the measure of angle OAB.

Solution

To find the measure of angle OAB, we can use the fact that the sum of the angles in a triangle is always 180 degrees.

Let's denote the measure of angle OAB as x. Since angle AOB is 90 degrees, angle ABO is also 90 degrees. Therefore, the sum of angles ABO and OAB is 180 - 90 = 90 degrees.

We also know that angle ACB is 45 degrees. Since angle ABO is 90 degrees, the sum of angles ABO and OBC is 180 - 45 = 135 degrees.

Now, let's consider triangle ABC. The sum of angles ABC, BCA, and CAB is 180 degrees. We can express angle ABC as the sum of angles ABO and OBC: ABC = ABO + OBC = 90 + 135 = 225 degrees.

Since the sum of angles in a triangle is always 180 degrees, we can write the equation: ABC + ACB + BCA = 180. Substituting the known values, we get: 225 + 45 + BCA = 180.

Simplifying the equation, we find: BCA = -90 degrees.

However, angles cannot have negative measures. Therefore, there is no solution for the measure of angle OAB that satisfies the given conditions.

Conclusion

Based on the given information, there is no solution for the measure of angle OAB that satisfies the conditions of the problem.