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.