A museum was designed without windows, but a truncated cylindrical skylight lightens the building. In the development of this, the architect has areas where the angles formed at junctures are vertical angles.

If a pair of vertical angles measures 5x - 54 and 3x -24, find the measures of the angles.

since vertical angles are equal,

5x-54 = 3x-24
2x = 30
x = 15

The angles are 21 degrees

Vertical angles are formed when two lines intersect each other. These angles are always congruent, which means they have the same measure.

Let's use the given expressions for the measures of the vertical angles:

Angle 1: 5x - 54
Angle 2: 3x - 24

Since vertical angles are congruent, we can set up an equation to find the value of x:

5x - 54 = 3x - 24

Now, let's solve this equation:

5x - 3x = -24 + 54
2x = 30
x = 15

Now that we have the value of x, we can substitute it back into the expressions for the angles to find their measures:

Angle 1: 5x - 54 = 5(15) - 54 = 75 - 54 = 21
Angle 2: 3x - 24 = 3(15) - 24 = 45 - 24 = 21

Therefore, the measures of the angles are both 21 degrees.

To find the measures of the vertical angles, we can set the two expressions equal to each other since vertical angles are congruent.

So, we have:

5x - 54 = 3x - 24

Now, we will solve this equation for x:

5x - 3x = -24 + 54
2x = 30
x = 30/2
x = 15

Now that we have the value of x, we can substitute it back into either expression to find the measures of the angles.

Let's use the first expression:

Angle 1 = 5x - 54 = 5(15) - 54 = 75 - 54 = 21

Therefore, the measure of the first angle is 21 degrees.

To find the measure of the second angle, we can substitute x = 15 into the second expression:

Angle 2 = 3x - 24 = 3(15) - 24 = 45 - 24 = 21

Therefore, the measure of the second angle is also 21 degrees.

So, both angles measure 21 degrees.