Let's analyze the problem and provide counterexamples for each statement as requested:

(a) If the measures of $\angle R$, $\angle S$, and $\angle T$ sum to $180^\circ$, then one of the angles must be obtuse.

A triangle's angles sum to $180^\circ$, but none of the angles have to be obtuse. For example:

• Counterexample: $m\angle R = 60^\circ$, $m\angle S = 60^\circ$, $m\angle T = 60^\circ$.

These angles form an equilateral triangle, where all angles are acute.

(b) If $m\angle WXZ = 60^\circ$, and point $Y$ is in the interior of $\angle WXZ$, then $m\angle WXY = 40^\circ$ and $m\angle YXZ = 20^\circ$.

The sum of $m\angle WXY$ and $m\angle YXZ$ must equal $m\angle WXZ = 60^\circ$. However, the values $40^\circ$ and $20^\circ$ provided do not necessarily hold true as unique possibilities. A counterexample is:

• Counterexample: $m\angle WXY = 30^\circ$, $m\angle YXZ = 30^\circ$.

Here, the angles still add up to $60^\circ$, but they differ from the given condition.

(c) If the area of a rectangle is $100$, then the length is $10$ and the width is $10$.

The area $A$ of a rectangle is calculated as $A = \text{length} \times \text{width}$. However, the length and width do not have to be $10$ each. For example:

• Counterexample: length = $20$, width = $5$.

Here, $20 \times 5 = 100$, which still satisfies the area condition.

(d) If $\angle 1$ and $\angle 2$ are supplementary angles, then one of them must be acute.

Supplementary angles sum to $180^\circ$, but neither angle has to be acute. For example:

• Counterexample: $m\angle 1 = 90^\circ$, $m\angle 2 = 90^\circ$.

Both angles are right angles, and neither is acute.

Summary of Counterexamples:

(a) $m\angle R = 60^\circ, m\angle S = 60^\circ, m\angle T = 60^\circ$
(b) $m\angle WXY = 30^\circ, m\angle YXZ = 30^\circ$
(c) Length = $20$, Width = $5$
(d) $m\angle 1 = 90^\circ, m\angle 2 = 90^\circ$

Do you need further clarification or a step-by-step breakdown of any specific part?
Here are 5 related questions you can explore:

1. What are the properties of angles in different types of triangles (acute, obtuse, right)?
2. How do you calculate the area of a rectangle with varying dimensions?
3. What are supplementary angles, and how are they related to linear pairs?
4. How do interior points of an angle split it into smaller angles?
5. Can two angles be supplementary if both are obtuse?

Tip: In geometry, counterexamples are powerful tools to disprove false statements by showing specific cases where the claims fail.