Let’s analyze the problem:

We are tasked with solving the following:

• $ABCD$ is a quadrilateral.
• $BO$ and $CO$ are the angle bisectors of $\angle B$ and $\angle C$, respectively.
• Given:
  • $\angle DAB = 60^\circ$,
  • $\angle ABC = 60^\circ$,
  • $\angle ADC = 100^\circ$,
• We need to find the measures of $\angle BOC$ and $\angle BCD$.

Step-by-step Solution:

Step 1: Sum of angles in a quadrilateral

The sum of interior angles of a quadrilateral is always $360^\circ$. Therefore: $\angle DAB + \angle ABC + \angle BCD + \angle ADC = 360^\circ$

Substituting the given values: $60^\circ + 60^\circ + \angle BCD + 100^\circ = 360^\circ$

Simplify: $\angle BCD = 360^\circ - 220^\circ = 140^\circ$

Step 2: Relation between $\angle BOC$ and $\angle B$ & $\angle C$

Since $BO$ and $CO$ are angle bisectors of $\angle B$ and $\angle C$, respectively, the measure of $\angle BOC$ is given by the formula: $\angle BOC = 180^\circ - \frac{\angle B + \angle C}{2}$

From the given: $\angle B = \angle ABC = 60^\circ$, $\angle C = \angle BCD = 140^\circ$.

Substitute these values: $\angle BOC = 180^\circ - \frac{60^\circ + 140^\circ}{2}$

Simplify: $\angle BOC = 180^\circ - \frac{200^\circ}{2} = 180^\circ - 100^\circ = 80^\circ$

Final Answers:

1. $\angle BCD = 140^\circ$
2. $\angle BOC = 80^\circ$

Do you want a more detailed explanation or further clarification?

Relative Questions:

1. What is the general formula for the sum of angles in a polygon?
2. How do angle bisectors divide an angle geometrically?
3. What is the relationship between opposite angles in a cyclic quadrilateral?
4. How can $\angle BOC$ be calculated if $ABCD$ is inscribed in a circle?
5. How does the type of quadrilateral (e.g., parallelogram, trapezium) affect angle calculations?

Tip:

Always check the type of quadrilateral (cyclic, parallelogram, etc.) to apply the correct geometric properties!