Question

Find the value of each variable in the circle to the right. The dot represents the center of the circle.

Answer 1

Angle d is 65°.

Angle c is 89°.

Arc a is 131°.

Arc b is 47°.

Explanation:

According to the inscribed quadrilateral theorem, opposite angles of the quadrilateral are supplementary, which means they sum 180°.

`115+d=180\\d=180-115\\d=65`

Therefore, angle d is 65°.

`c+91=180\\c=180-91\\c=89`

Therefore, angle c is 89°.

Now, the angle 115° subtends the arc 99+a, which according to the theorem

`115=(1)/(2)(99+a)\\ 230=99+a\\a=230-99\\a=131`

Therefore, arc a is 131°.

Similarly, angle c subtends arc a+b, which means

`c=(1)/(2)(a+b)\\ 89=(1)/(2)(131+b)\\ 178=131+b\\b=178-131\\b=47`

Therefore, arc b is 47°.