Question

Луч OV биссектриса ∠ZOY, а луч OU биссектриса ∠YOX. Найти ∠ZOX если ∠VOU=80°.

Answer 1

<XOZ=<ZOV+<VOY+<YOU+<UOX
<ZOV=<VOY - VO бисс.
<YOU=<UOX - UO бисс. 
<UOV=<VOY+<YOU=80
=> <XOZ=80+80=160

Answer 2

Problem Statement

We are given that the ray OV is the bisector of angle ZOY, and the ray OU is the bisector of angle YOX. We need to find the measure of angle ZOX if angle VOU is 80°.

Solution

To find the measure of angle ZOX, we can use the given information and apply the angle bisector theorem.

According to the angle bisector theorem, if a ray bisects an angle, it divides the opposite side into two segments that are proportional to the adjacent sides of the angle.

In this case, the ray OV bisects angle ZOY, so it divides the opposite side OY into two segments, ZO and OY, that are proportional to the adjacent sides ZV and VY.

Similarly, the ray OU bisects angle YOX, so it divides the opposite side OX into two segments, YO and OX, that are proportional to the adjacent sides YU and UX.

Let's denote the lengths of the segments as follows: - ZO = a - OY = b - YO = c - OX = d

Using the angle bisector theorem, we can set up the following proportions: 1. For ray OV: - ZO/OY = ZV/VY - a/b = ZV/VY

2. For ray OU: - YO/OX = YU/UX - c/d = YU/UX

We are given that angle VOU is 80°. Since OV and OU are bisectors, we know that angles ZOV and YOU are equal. Therefore, we have: - angle ZOV = angle YOU = 80°

Now, let's solve for the measure of angle ZOX.

Since OV is the bisector of angle ZOY, we can write: - ZV/VY = a/b

Similarly, since OU is the bisector of angle YOX, we can write: - YU/UX = c/d

Since angles ZOV and YOU are equal, we can write: - ZV/VY = YU/UX

Combining the above equations, we have: - a/b = c/d

Now, let's solve for the measure of angle ZOX.

Since angle ZOX is the sum of angles ZOV and YOU, we have: - angle ZOX = angle ZOV + angle YOU = 80° + 80° = 160°

Therefore, the measure of angle ZOX is 160°.

Conclusion

In conclusion, given that the ray OV is the bisector of angle ZOY, the ray OU is the bisector of angle YOX, and angle VOU is 80°, we can determine that the measure of angle ZOX is 160°.