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.