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To solve for the unknown angles in the provided quadrilaterals, let's analyze each one using the fact that the sum of the interior angles in any quadrilateral is \( 360^\circ \).

### Quadrilateral **a**:
Given angles:
- \( 81^\circ \)
- \( 77^\circ \)
- \( x^\circ \)
- The fourth angle (not labeled, but complementary to \( x^\circ \)).

### Quadrilateral **b**:
Given angles:
- \( 117^\circ \)
- \( 85^\circ \)
- \( b^\circ \)
- \( c^\circ \)

Let's set up the equations for each quadrilateral.

For **Quadrilateral a**:
\[
81^\circ + 77^\circ + x^\circ + (\text{fourth angle}) = 360^\circ
\]
The fourth angle can be found by using the trapezoid properties or directly calculating from the equation.

For **Quadrilateral b**:
\[
117^\circ + 85^\circ + b^\circ + c^\circ = 360^\circ
\]

Now, I'll solve for \( x \), \( b \), and \( c \).Here are the solutions for the unknown angles in each quadrilateral:

### Quadrilateral **a**:
- \( x = 202^\circ - \text{fourth angle} \)
- The fourth angle is simply the complementary angle to \( x \).

### Quadrilateral **b**:
- \( b = 158^\circ - c \)
- \( c \) remains as an unknown, meaning \( b \) and \( c \) are dependent on each other.

In Quadrilateral **a**, the exact value of \( x \) depends on the fourth angle. For Quadrilateral **b**, the angles \( b \) and \( c \) are complementary to each other in the sense that they sum to \( 158^\circ \).

Would you like further clarification, or do you have any other questions?

Here are 5 related questions to deepen your understanding:

1. How would you find the exact values of \( b \) and \( c \) if one was provided?
2. What properties of trapezoids can help simplify finding angles?
3. How can the exterior angles of a polygon help in determining interior angles?
4. How would the situation change if these quadrilaterals were cyclic (inscribed in a circle)?
5. How does the knowledge of parallel sides in quadrilaterals aid in finding angles?

**Tip:** In many geometric problems, knowing properties such as supplementary, complementary angles, and specific quadrilateral types (like trapezoids or parallelograms) can make it easier to solve for unknown values.

Learn how to solve for unknown angles in quadrilaterals using the sum of interior angles theorem. Detailed step-by-step solutions for Quadrilateral a and Quadrilateral b. Suitable for high school students studying geometry and angle properties.

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