Arc XYZ lies on a circle centered at O. Is a < 55?

VERITAS PREP OFFICIAL SOLUTION:

The correct response is (C).

If we draw a radius from O to Y, we can fill in the two given values: ZOY = 35, and XOY = 75. OX and OZ are both radii of the same circle, so their opposite angles must be equal. We can use this information to solve for angle XYZ, and in term find the value for a.

To do the math, here’s how it would look:



180 = 35 + OYZ + OZY
145/2 = 72.5 = OYZ = OZY

180 = 75 + OXY + OYX
105/2 = 52.5 = OXY + OYX

Angle XYZ = 72.5 + 52.5 = 125

Angle a = 180 – 125
Angle a = 55

The answer to the question would always be “no.”

If you chose (A), this would allow us to solve for XYO, but we’d also need angle OYZ to find a.

If you chose (B), this would allow us to find OYZ, but we’d also need to know angle XOY to find angle XYO.

If you chose (D), both pieces of information are important, but neither is independently sufficient.

If you chose (E), you may have missed that XO = OY = OZ, since all radii in a circle must be equal, as well as the fact that angles opposite equal sides are also equal.