Analyzing the given information:

Let's break down the information provided in the problem statement:

1. Angle A is equal to the sum of the other two angles.
2. The ratio of angle B to angle C is 4:5.

From this information, we can deduce that angle A must be greater than angles B and C since it is equal to their sum. Additionally, the sum of angles B and C must be less than 180 degrees, as it is not possible for a triangle to have angles that add up to more than 180 degrees.

Determining the values of the angles:

Let's assign variables to the angles to make it easier to work with:

Let angle A be x degrees.
Let angle B be 4y degrees.
Let angle C be 5y degrees.

According to the first piece of information, we know that:

x = (4y + 5y)

Simplifying the equation:

x = 9y

Now, we have the value of angle A in terms of y.

Applying the sum of angles in a triangle:

The sum of angles in a triangle is always 180 degrees. Therefore, we can write the equation:

x + 4y + 5y = 180

Substituting the value of x from the previous equation:

9y + 4y + 5y = 180

Combining like terms:

18y = 180

Dividing both sides by 18:

y = 10

Now that we have the value of y, we can substitute it back into our earlier equations to find the values of x, B, and C.

Calculating the angles:

Substituting y = 10 into the equations:

x = 9y = 9 * 10 = 90 degrees
B = 4y = 4 * 10 = 40 degrees
C = 5y = 5 * 10 = 50 degrees

Therefore, the measure of angle A is 90 degrees, angle B is 40 degrees, and angle C is 50 degrees.

Conclusion:

In summary, based on the given information, we determined that angle A is equal to the sum of angles B and C. By assigning variables to the angles, we were able to solve for their values. The ratio of angle B to angle C provided additional clues to help us calculate the specific measures. The final solution is that angle A is 90 degrees, angle B is 40 degrees, and angle C is 50 degrees.

Angle A =90 ,B = 40,C=50