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Let's solve the problems step by step.

### 1. Parallelogram \( ABCD \):
We are asked to find the values of \( x \), \( y \), and \( z \) for this parallelogram.

- **Angle properties of a parallelogram**: Opposite angles are equal, and adjacent angles are supplementary (sum to 180°).

Given that \(\angle B = 100^\circ\), we can deduce the following:
- \( \angle D = \angle B = 100^\circ \) (since opposite angles in a parallelogram are equal). Therefore, \( y = 100^\circ \).
- \( \angle A + \angle B = 180^\circ \) (adjacent angles in a parallelogram are supplementary), so:
  \[
  \angle A = 180^\circ - 100^\circ = 80^\circ
  \]
  Thus, \( z = 80^\circ \).

Now, because \( \angle A = \angle C \) (opposite angles), we have:
- \( x = 80^\circ \).

**Summary of results**:
- \( x = 80^\circ \)
- \( y = 100^\circ \)
- \( z = 80^\circ \)

---

### 2. Parallelograms \( GUNS \) and \( RUNS \):

#### (i) Parallelogram \( GUNS \):
We are given:
- \( GU = 3y - 1 \) and \( NS = 18 \)
- \( GS = 3x \) and \( UN = 26 \)

Since opposite sides of a parallelogram are equal:
- \( GU = NS \), so:
  \[
  3y - 1 = 18
  \]
  Solving for \( y \):
  \[
  3y = 19 \quad \Rightarrow \quad y = \frac{19}{3} \approx 6.33
  \]
- \( GS = UN \), so:
  \[
  3x = 26 \quad \Rightarrow \quad x = \frac{26}{3} \approx 8.67
  \]

**Results for (i)**:
- \( x = 8.67 \) cm
- \( y = 6.33 \) cm

#### (ii) Parallelogram \( RUNS \):
We are given:
- \( RO = r + y \), \( ON = y + 7 \)
- \( RU = 20 \), \( SU = 16 \)

Since diagonals of a parallelogram bisect each other, the diagonals divide into equal parts, so:
- \( RO = ON \), implying:
  \[
  r + y = y + 7
  \]
  Simplifying, we get:
  \[
  r = 7
  \]

Since this is the only unknown provided, the value of \( r \) is \( 7 \).

**Result for (ii)**:
- \( r = 7 \)

---

### Summary of Solutions:
1. \( x = 80^\circ \), \( y = 100^\circ \), \( z = 80^\circ \)
2. (i) \( x \approx 8.67 \) cm, \( y \approx 6.33 \) cm
   (ii) \( r = 7 \)

Would you like more details or clarification on any step? 

Here are some related questions to further your understanding:
1. What are the properties of diagonals in a parallelogram?
2. How do opposite angles in a parallelogram relate to each other?
3. Why are adjacent angles in a parallelogram always supplementary?
4. How do we know that diagonals bisect each other in a parallelogram?
5. What other quadrilaterals share properties with parallelograms?

**Tip**: In a parallelogram, opposite sides are always equal in length, and diagonals bisect each other, making calculations easier when finding unknowns.

1. Consider the following parallelogram. Find the values of the unknowns x, y, z. 2. The following figures GUNS and RUNS are parallelograms. Find x and y (Lengths are in cm).

Learn how to solve for unknowns in parallelograms by applying properties of angles, diagonals, and side lengths. This problem focuses on angles in a parallelogram and the equality of opposite sides, suitable for Grades 7-9 geometry.

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