Final answer:
To find the value of ∠AOK, we need to use the given information that AB≅BC and AO≅OC, and that OK is the angle bisector of ∠BOC. Using the properties of isosceles triangles and the fact that OK bisects the angle BOC, we can determine that the measure of ∠AOK is equal to 90 degrees minus the measure of ∠ABC. Therefore, m∠AOK = 90 - m∠ABC. From the given options, m∠AOK is equal to 30 degrees, so the correct answer is D.
Step-by-step explanation:
To find the value of ∠AOK, we need to use the given information that AB ≅ BC and AO ≅ OC, and that OK is the angle bisector of ∠BOC. Since AB ≅ BC, triangle ABC is an isosceles triangle. Therefore, ∠ABC and ∠ACB are equal. Similarly, since AO ≅ OC, triangle AOC is also an isosceles triangle. Therefore, ∠AOC and ∠ACO are equal.
Since OK is the angle bisector of ∠BOC, it divides the angle into two equal parts. Therefore, ∠AOK = ∠AOC / 2. Since ∠AOC and ∠ACO are equal from the isosceles triangles, we have ∠AOC = ∠ACO. Therefore, ∠AOK = ∠AOC / 2 = ∠ACO / 2.
Since triangle ABC is isosceles, we have ∠ACO = ∠ABC from alternate interior angles. Therefore, ∠AOK = ∠ABC / 2. However, since ∠ABC = ∠ACB, we can also say that ∠AOK = ∠ACB / 2. Since ∠ACB is equal to 180 - ∠ABC - ∠ABC (because the angles in a triangle add up to 180 degrees), we can substitute this into the equation to get ∠AOK = (180 - ∠ABC - ∠ABC) / 2 = (180 - 2∠ABC) / 2 = 90 - ∠ABC.
Therefore, m∠AOK = 90 - m∠ABC. We know that AB ≅ BC, so ∠ABC is an isosceles triangle and m∠ABC = m∠ACB. From the given options, m∠AOK would be equal to 90 degrees minus the value of m∠ABC. Therefore, the correct answer is D. m∠AOK = 30 degrees.