find the value of each variable. for the circle, the dot represents the center.\n$a = \\square$ (simplify…
find the value of each variable. for the circle, the dot represents the center.\n$a = \\square$ (simplify your answer.)
Answer
Explanation:
Step1: Identify the inscribed quadrilateral property
Opposite angles of a cyclic quadrilateral are supplementary. $$c^{\circ} + 94^{\circ} = 180^{\circ}$$ $$d^{\circ} + 117^{\circ} = 180^{\circ}$$
Step2: Calculate the value of $c$
$$c = 180 - 94 = 86$$
Step3: Calculate the value of $d$
$$d = 180 - 117 = 63$$
Step4: Use the inscribed angle theorem for $a$
An inscribed angle is half the intercepted arc. $$c^{\circ} = \frac{1}{2}(a^{\circ} + b^{\circ})$$ $$86 = \frac{1}{2}(a + b)$$
Step5: Use the inscribed angle theorem for $d$
$$d^{\circ} = \frac{1}{2}(b^{\circ} + 94^{\circ})$$ $$63 = \frac{1}{2}(b + 94)$$
Step6: Solve for $b$
$$126 = b + 94$$ $$b = 32$$
Step7: Solve for $a$
Substitute $b = 32$ into the equation from Step 4. $$172 = a + 32$$ $$a = 140$$
Answer:
a = 140