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1 of 81 of 8 Items Question Triangle XYZ is inscribed inside a circle with diameter XZ and the inscribed angle at Y. The angle measure at X is 59°. Using what you know about inscribed angles, find the arc measure of XY.(1 point) Responses 25° 25° 62° 62° 87° 87° Skip to navigation page 1 of 1

1 answer

To solve this problem, we need to understand the properties of inscribed angles and the arcs they subtend.

Since triangle XYZ is inscribed in a circle with diameter XZ, we know that angle Y is an inscribed angle that subtends arc XZ. The measure of an inscribed angle is half the measure of the arc it subtends.

Given that the angle at X measures 59°, we can use the following relationship:

  1. The measure of angle Y, which is inscribed, subtending arc XZ, is half the measure of arc XZ.
  2. Triangle XYZ's angles add up to 180°. Therefore, we can find the measure of angle Z:
    • Angle Z = 180° - Angle X - Angle Y
    • Angle Z = 180° - 59° - Angle Y.

However, we don't need angle Y directly since we need to find the arc XY instead.

The angle at X (59°) subtends arc YZ, which is opposite it. This means:

  • Arc XY + Arc YZ + Arc ZX = 360°.

Since arc YZ corresponds to angle X (59°), we can say:

  • Measure of arc YZ = 2 * Angle X = 2 * 59° = 118°.

Now let's denote the measure of arc XY as A and arc ZX as B. We know:

  • A + 118° + B = 360°.

Since angle Z is opposite arc XY and contributes to the equation of the total circle, we can denote it as A = 2 * measure of angle Y.

However, we don't specifically have angle Y unless we know angle Z's measure.

But it says we are looking for arc XY. The relationship we can derive from arc YZ is:

The sum of arcs around the circle equals 360°. Since we recognized that the opposite angles in triangle XY would still apply, we can calculate indirectly.

Assuming we have only one angle (59°) at X, we can point that the total arc must average out to total the remaining angle that would satisfy circle's arc properties.

Thus we can reasonably deduce that if angle X subtended arc YZ, which as computed is 118°, then for it to add up:

Arc XY must be less than thus working towards our circle closure which would yield:

  • Arc XY = 360° - 118° = 242°/2 = 121°.

Notice that the arc XY should be the outer subtended area. We have potential arc YZ influence already considered.

However, to find arc XY we realize the net sum for the circle must hold by previous breakdown:

So confirming given response options, the calculation directly leads to:

  • Arc measure of XY = 180 - 118 (direct opposite angle on circle boundary influence!) = 62°.

Thus,

The answer is 62°.