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### Problem 3: Given that \( AB \parallel DC \), find the values of the unknowns.

#### Part (a)
- Figure: A trapezoid with parallel sides \( AB \parallel DC \).
- Unknown: \( a \).

The sum of the angles on the same side of a transversal line between parallel lines must be \( 180^\circ \). So, in trapezoid \( ABCD \):

\[
\text{Angle } D + \text{Angle } A = 180^\circ
\]

Given that angle \( D \) is \( 70^\circ \):

\[
a = 180^\circ - 70^\circ = 110^\circ
\]

#### Part (b)
- Figure: A quadrilateral \( ABCD \) with diagonals.
- Unknowns: \( b' \), \( b \), and \( d' \).

Since the diagonals of the quadrilateral bisect each other (in a parallelogram):

- \( b' = b = 76^\circ \) because opposite angles are equal.
- To find \( d' \), note that \( d' + 118^\circ = 180^\circ \) (linear pair).
  
Thus:
\[
d' = 180^\circ - 118^\circ = 62^\circ
\]

#### Part (c)
- Figure: An isosceles triangle.
- Given angle: \( 58^\circ \) at vertex \( A \).

Since the triangle is isosceles, the two base angles (at \( B \) and \( C \)) are equal. Let's denote these angles as \( x \). The sum of angles in any triangle is \( 180^\circ \):

\[
x + x + 58^\circ = 180^\circ
\]

This simplifies to:

\[
2x = 180^\circ - 58^\circ = 122^\circ
\]

\[
x = \frac{122^\circ}{2} = 61^\circ
\]

Thus, both base angles are \( 61^\circ \).

### Problem 4: Find the values of the unknowns.

#### Part (a)
- Figure: A quadrilateral \( ABCD \).
- Given angles: \( \angle A = 114^\circ \), \( \angle B = 112^\circ \), \( \angle D = 95^\circ \).
- Unknown angle \( C \).

The sum of the interior angles of a quadrilateral is \( 360^\circ \):

\[
114^\circ + 112^\circ + 95^\circ + C = 360^\circ
\]

\[
C = 360^\circ - (114^\circ + 112^\circ + 95^\circ) = 360^\circ - 321^\circ = 39^\circ
\]

Thus, \( C = 39^\circ \).

#### Part (b)
- Figure: A quadrilateral with additional construction.
- Unknowns: \( b' \), \( b \), and angle \( C \).

Given:
- \( b = b' = 26^\circ \) because they are corresponding angles.

For angle \( C \), use the linear pair relationship where the sum of angles along a straight line is \( 180^\circ \). Given \( C + 34^\circ = 180^\circ \):

\[
C = 180^\circ - 34^\circ = 146^\circ
\]

### Summary of Results:
1. Part 3(a): \( a = 110^\circ \)
2. Part 3(b): \( b = b' = 76^\circ \), \( d' = 62^\circ \)
3. Part 3(c): \( \text{Base angles} = 61^\circ \)
4. Part 4(a): \( C = 39^\circ \)
5. Part 4(b): \( b = b' = 26^\circ \), \( C = 146^\circ \)

Would you like any further details or clarification on any part of the solution?

### Follow-up Questions:
1. What is the sum of the interior angles of a quadrilateral?
2. How do parallel lines affect the angles formed with transversals?
3. What is the property of angles in an isosceles triangle?
4. How do you calculate the unknown angle in a polygon?
5. What is the relationship between linear pairs of angles?

### Tip:
For quadrilaterals, always remember that the sum of the interior angles is \( 360^\circ \), while for triangles, it is \( 180^\circ \).

Find the value of unknown angles in the provided geometric figures.

This solution walks through solving for unknown angles in a variety of geometric figures including parallelograms, trapezoids, and triangles. Concepts covered include parallel lines and transversals, the interior angle sum theorem, and linear pair relationships.

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