Question

Дано: СО=DO; угол С=90 градусов; угол D=90 градусов Доказать: АО=ВО

Answer 1

Рассмотрим ΔAOD и ΔCOB:
 ∠AOD=∠COB как вертикальные, CO=OD, ∠C=∠D по условию. 
Следовательно ΔAOD=ΔCOB по второму признаку равенства треугольников, против равных углов лежат равные стороны, значит AO=OB,  что и требовалось доказать.

Answer 2

Given Information:

- SO = DO - Angle C = 90 degrees - Angle D = 90 degrees

To Prove:

- AO = BO

Solution:

To prove that AO = BO, we need to show that triangle AOB is an isosceles triangle, where AO and BO are the equal sides.

Since SO = DO, we can conclude that triangle SOD is an isosceles triangle, where SO and DO are the equal sides.

Since angle C = 90 degrees, we can conclude that triangle SOD is a right triangle.

Using the properties of right triangles, we can determine that angle SOD is also 90 degrees.

Now, let's consider triangle AOD. Since angle D = 90 degrees, we can conclude that triangle AOD is also a right triangle.

Since triangle AOD is a right triangle, we can use the Pythagorean theorem to find the relationship between the sides AO, OD, and AD.

According to the Pythagorean theorem, the square of the hypotenuse (AD) is equal to the sum of the squares of the other two sides (AO and OD).

Applying the Pythagorean theorem to triangle AOD, we have:

AO^2 + OD^2 = AD^2

Since angle SOD is 90 degrees, we can conclude that AD is equal to the hypotenuse of triangle SOD, which is SO or DO.

Therefore, we can rewrite the equation as:

AO^2 + OD^2 = SO^2

Since SO = DO, we can substitute DO for SO:

AO^2 + OD^2 = DO^2

Since angle D = 90 degrees, we can conclude that triangle BOD is also a right triangle.

Using the same reasoning as before, we can apply the Pythagorean theorem to triangle BOD:

BO^2 + OD^2 = DO^2

Since DO^2 = AO^2 + OD^2 (as shown above), we can substitute AO^2 + OD^2 for DO^2 in the equation for triangle BOD:

BO^2 + OD^2 = AO^2 + OD^2

Simplifying the equation, we get:

BO^2 = AO^2

Taking the square root of both sides, we have:

BO = AO

Therefore, we have proven that AO = BO, which means that triangle AOB is an isosceles triangle.

Thus, we have successfully demonstrated that AO = BO.

Note: The proof provided above is based on the given information and the properties of right triangles.