Question

given ⊙o below, if ⌢xy and ⌢yz are congruent, what is the measure of ∠yoz? a. 280° b. 80° c. 100° d. 140°

Explanation:

Step1: Recall circle - angle property

The sum of the central angles of a circle is 360°.

Step2: Let \(\angle XOY=\angle YOZ = x\)

Since \(\widehat{XY}\) and \(\widehat{YZ}\) are congruent, their central - angles are equal. Let \(\angle XOY = \angle YOZ=x\), and we know \(\angle XOZ = 80^{\circ}\).

Step3: Set up equation

\(\angle XOY+\angle YOZ+\angle XOZ=360^{\circ}\), substituting the values we get \(x + x+80^{\circ}=360^{\circ}\).

Step4: Simplify the equation

Combining like terms, \(2x+80^{\circ}=360^{\circ}\). Subtract 80° from both sides: \(2x=360^{\circ}-80^{\circ}=280^{\circ}\).

Step5: Solve for \(x\)

Divide both sides of the equation \(2x = 280^{\circ}\) by 2: \(x=\frac{280^{\circ}}{2}=140^{\circ}\). So \(\angle YOZ = 140^{\circ}\).

Answer:

D. \(140^{\circ}\)