Question

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

Answer 1

Answer:

Angle d is 65°.

Angle c is 89°.

Arc a is 131°.

Arc b is 47°.

Step-by-step 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

$115=\frac{1}{2}(99+a)\\ 230=99+a\\a=230-99\\a=131$

Therefore, arc a is 131°.

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

$c=\frac{1}{2}(a+b)\\ 89=\frac{1}{2}(131+b)\\ 178=131+b\\b=178-131\\b=47$

Therefore, arc b is 47°.