Question

В прямоугольном треугольнике проведена высота из вершины прямого угла. Докажите, что данный треугольник и два образовавшихся треугольника имеют соответственно равные углы. Помогите пожалуйста, как решить эту задачу?

Answer 1

Proof that a Right Triangle and Its Two Resulting Triangles Have Corresponding Equal Angles

To prove that a right triangle and the two resulting triangles formed by drawing an altitude from the vertex of the right angle have corresponding equal angles, we can use the properties of similar triangles.

Let's consider a right triangle ABC, where angle B is the right angle. We draw an altitude from vertex B to side AC, creating two resulting triangles: ABP and CBP.

To prove that these triangles have corresponding equal angles, we will show that they are similar triangles.

Step 1: Show that triangle ABP is similar to triangle ABC.

Since angle B is a right angle, angle ABP is also a right angle. Therefore, triangle ABP is a right triangle.

We can prove that triangle ABP is similar to triangle ABC by showing that they share one equal angle and the ratios of their corresponding sides are equal.

- Angle A is common to both triangles. - The ratio of the lengths of the corresponding sides is BP/BC.

Therefore, triangle ABP is similar to triangle ABC by the Angle-Angle (AA) similarity criterion.

Step 2: Show that triangle CBP is similar to triangle ABC.

Since angle B is a right angle, angle CBP is also a right angle. Therefore, triangle CBP is a right triangle.

We can prove that triangle CBP is similar to triangle ABC by showing that they share one equal angle and the ratios of their corresponding sides are equal.

- Angle C is common to both triangles. - The ratio of the lengths of the corresponding sides is BP/BA.

Therefore, triangle CBP is similar to triangle ABC by the Angle-Angle (AA) similarity criterion.

Conclusion:

From the above proof,