ray OB is the bisector of angle AOC and ray OC is the bisector of angle BOD. m<AOD =120 find m<AOB
Since OB bisects AOC, AOB=BOC
Since OC bisects BOD, BOC=COD
So, all 3 angles are equal, and are 120/3 = 40
yeah, I know, the terminology is loose, but you can fill in the appropriate correct language
To find the measure of angle AOB, we need to use the fact that ray OB is the bisector of angle AOC and ray OC is the bisector of angle BOD.
Let's start by labeling the angles:
m<AOC = x (since ray OB is the bisector of angle AOC)
m<BOC = x (since ray OC is the bisector of angle BOD)
m<AOD = 120 (given)
Since ray OB is the bisector of angle AOC, we can deduce that angles AOB and BOC are vertical angles, and thus they are congruent. Similarly, ray OC bisects angle BOD, so angles AOC and DOB are also congruent.
Using these facts, we can write the following equations:
m<AOB = m<BOC (vertical angles)
m<AOC = m<DOC (angle bisector)
m<AOC + m<BOC + m<DOC = 180 (angle sum of a triangle)
Substituting the known values and the congruent angles into the equation:
x + x + 120 = 180
2x + 120 = 180
2x = 60
x = 30
So, the measure of angle AOB is m<AOB = m<BOC = x = 30 degrees.
To find the measure of angle AOB, we can first find the measure of angle COD, which is supplementary to angle AOD.
Since ray OB is the bisector of angle AOC, it divides the angle into two equal parts: angle AOB and angle BOC. Let's call the measure of angle AOB as x. Therefore, the measure of angle BOC is also x.
Similarly, since ray OC is the bisector of angle BOD, it divides the angle into two equal parts: angle BOC and angle COD. Since BOC is equal to x, angle COD is also equal to x.
Now, we know that the sum of angle measures in a straight line (180 degrees) is equal to the supplementary angle AOD, which is given as 120 degrees.
Therefore, we can write the equation: x + x + x = 120
Simplifying the equation, we get: 3x = 120
Dividing both sides of the equation by 3, we find: x = 40
Hence, the measure of angle AOB is 40 degrees.