The measure of arc ST and angle CED in question 5 and 6 are 90 degrees and 67 degrees respectively.
The figures in the image is a circle with an inscribed angle and an arc.
To solve for the measure of arc ST and angle CED, first we solve for the value of x in each circle using the angle-arc relationship.
Note that:
Inscribed angle = 1/2 × Intercepted arc
In question 5)
Inscribed angle = 4x - 15
Intercepted arc ST = 5x + 15
Plug these values into the above formula and solve for x:
Inscribed angle = 1/2 × Intercepted arc
4x - 15 = 1/2 × ( 5x + 15 )
2( 4x - 15 ) = 5x + 15
8x - 30 = 5x + 15
8x - 5x = 15 + 30
3x = 45
x = 45/3
x = 15
Now, the measure of arc ST will be:
Intercepted arc ST = 5x + 15
Plug in x = 15:
Intercepted arc ST = 5(15) + 15
Intercepted arc ST = 75 + 15
Intercepted arc ST = 90 degrees
In question 6)
Inscribed angle = 15x + 7
Intercepted arc DC = 33x + 2
Find x using the angle-arc relationship:
Inscribed angle = 1/2 × Intercepted arc
15x + 7 = 1/2 × ( 33x + 2 )
2( 15x + 7 ) = 33x + 2
30x + 14 = 33x + 2
33x - 30x = 14 - 2
3x = 12
x = 12/3
x = 4
Now, the measure of angle CED will be:
Angle CED = 15x + 7
Plug in x = 4:
Angle CED = 15(4) + 7
Angle CED = 60 + 7
Angle CED = 67 degrees.
Therefore, the measure of angle CED is 67 degrees.