Question

Два угла треугольника равны 58° и 72°. Найдите тупой угол, который образуют высоты треугольника, выходящие из вершин этих углов. Ответ дайте в градусах. (2 способа решения)

Answer 1

Дан тр-ник АВС. угол А = 58, угол В = 72 градуса. АН и ВД - высоты. О - точка пересечения высот. Найти угол АОВ.
Решение.
Рассмотрим прямоугольный тр-ник АНВ. Так как угол В = 72, то угол ВАН = 90 - 72 = 18 градусов.
Рассмотрим прямоугольный тр-ник АДВ. Так как угол А = 58, то угол АВД = 90 - 58 = 32 градуса.
Рассмотрим тр-ник АОВ, в котором уже известны два угла: 18 и 38 градусов.
Искомый угол АОВ = 180 - (18 + 32) = 120 градусов.
Ответ: 120.

Answer 2

Problem Analysis

We are given two angles of a triangle, 58° and 72°. We need to find the obtuse angle formed by the altitudes of the triangle coming out of these angles.

Solution 1: Using Angle Sum Property of a Triangle

In a triangle, the sum of all angles is always 180°. Therefore, we can find the third angle of the triangle by subtracting the sum of the given angles from 180°.

Let's calculate the third angle: - Given angle 1: 58° - Given angle 2: 72°

Using the angle sum property: Third angle = 180° - (angle 1 + angle 2)

Substituting the given values: Third angle = 180° - (58° + 72°)

Now we can calculate the obtuse angle formed by the altitudes of the triangle coming out of these angles.

Solution 2: Using Complementary Angles

We know that the sum of an angle and its complement is always 90°. Therefore, we can find the complement of each given angle and then calculate the obtuse angle formed by the altitudes.

Let's calculate the complements of the given angles: - Given angle 1: 58° - Given angle 2: 72°

Complement of angle 1 = 90° - angle 1 Complement of angle 2 = 90° - angle 2

Now we can calculate the obtuse angle formed by the altitudes of the triangle coming out of these angles.

Calculation

Let's calculate the third angle using both methods:

Solution 1: Using Angle Sum Property of a Triangle Third angle = 180° - (58° + 72°) Third angle = 180° - 130° Third angle = 50°

Solution 2: Using Complementary Angles Complement of angle 1 = 90° - 58° Complement of angle 1 = 32°

Complement of angle 2 = 90° - 72° Complement of angle 2 = 18°

Now we can calculate the obtuse angle formed by the altitudes of the triangle coming out of these angles.

Obtuse Angle Calculation

To find the obtuse angle formed by the altitudes of the triangle, we need to subtract the sum of the complements from 180°.

Obtuse angle = 180° - (complement of angle 1 + complement of angle 2) Obtuse angle = 180° - (32° + 18°) Obtuse angle = 180° - 50° Obtuse angle = 130°

Therefore, the obtuse angle formed by the altitudes of the triangle, coming out of the given angles, is 130°.

Note: The two solutions provided above yield the same result, which confirms the accuracy of the answer.