The image presents a geometry problem involving angles and a triangle in a coordinate plane. Let's analyze and solve the problem step by step.

Problem Breakdown:

The question involves a right triangle $\triangle OAB$, with the right angle located at $O$, and vertex $O$ positioned on a straight line $CD$. A line $OE$ bisects $\angle AOD$, and the question presents two scenarios:

1. For Figure ①: When $B$ is on the same side of $CD$ as $A$, and $\angle AOC = 58^\circ$, find $\angle BOE$.

2. For Figure ②: When $B$ is on the opposite side of $CD$ relative to $A$, and $\angle AOE = 2 \angle BOD$, find $\angle AOC$.

Step 1: Figure ① Analysis

• Given:

  • $\angle AOC = 58^\circ$.
  • $OE$ is the angle bisector of $\angle AOD$.

• Steps:

  1. Since $OE$ bisects $\angle AOD$, the measure of $\angle BOE$ can be determined as: $\angle AOD = \angle AOC + \angle COD = 58^\circ + \angle COD$
  2. Let $\angle COD = x$. Then $\angle AOD = 58^\circ + x$.
  3. Since $OE$ bisects $\angle AOD$: $\angle BOE = \frac{1}{2} \times \angle AOD = \frac{1}{2} \times (58^\circ + x).$

The value of $x$ (or $\angle COD$) must be provided or deduced from the problem setup.

Step 2: Figure ② Analysis

• Given:

  • $\angle AOE = 2 \angle BOD$.
  • $OE$ is the angle bisector of $\angle AOD$.

• Steps:

  1. Since $OE$ bisects $\angle AOD$, we write: $\angle AOD = \angle AOE + \angle BOD.$
  2. Substituting $\angle AOE = 2 \angle BOD$ into the equation: $\angle AOD = 2 \angle BOD + \angle BOD = 3 \angle BOD.$
  3. Let $\angle AOD = x$, then $x = 3 \angle BOD$, so: $\angle BOD = \frac{x}{3}.$
  4. Since $OE$ bisects $\angle AOD$, we also have: $\angle AOC = \angle AOD / 2 = x / 2.$

Thus, $\angle AOC$ depends on determining $x$, which can be computed based on additional constraints or the relationships among the angles.

Request for Clarification:

1. Do you have additional information about $\angle COD$ in Figure ①?
2. Should I proceed to compute symbolic solutions assuming $\angle COD$ and $\angle BOD$ are unknowns?
3. Would you like detailed drawings or geometric proofs to accompany the solutions?

Follow-Up Questions:

1. What is the relationship between $\angle COD$ and the other angles in Figure ①?
2. Can $\angle AOC$ in Figure ② be expressed in terms of other known angles or proportions?
3. What are the implications of $B$ being on the same side vs. the opposite side in terms of angle measures?
4. How does the line $OE$ splitting $\angle AOD$ influence the symmetry of the setup?
5. Are the angles $\angle AOD$, $\angle BOE$, or other elements tied to any external geometric constraints?

Tip:

When solving angle problems involving bisectors, always express unknown angles in terms of a single variable to simplify calculations.