Let's go through each of the statements and provide counterexamples as requested:

(a) Statement: "If the measures of ∠P, ∠Q, and ∠R sum to 180°, then all of the angles must be acute."

• Explanation: This statement is false because angles that sum to 180° can include right or obtuse angles. For example, a right triangle has one right angle (90°), and the other two are acute.
• Counterexample: $m∠P = 90^\circ$, $m∠Q = 60^\circ$, and $m∠R = 30^\circ$. These sum to 180°, but ∠P is not acute (it's a right angle).

(b) Statement: "If ∠1 and ∠2 are supplementary angles, then one of them must be obtuse."

• Explanation: This statement is false because supplementary angles can both be acute, as long as their sum is 180°. For example, they can both be 90° each (right angles).
• Counterexample: $m∠1 = 90^\circ$, $m∠2 = 90^\circ$. Both are right angles, and neither is obtuse.

(c) Statement: "If the area of a rectangle is 16, then the length is 4 and the width is 4."

• Explanation: This is false because the dimensions of the rectangle could vary as long as the product of the length and width equals 16. For example, the length could be 8 and the width 2.
• Counterexample: Length = 8, Width = 2.

(d) Statement: "If $m∠ABD = 30^\circ$, and point C is in the interior of ∠ABD, then $m∠ABC = 18^\circ$ and $m∠CBD = 12^\circ$."

• Explanation: This is false because the division of ∠ABD into two smaller angles is not necessarily 18° and 12°. They could be divided differently.
• Counterexample: $m∠ABC = 20^\circ$, $m∠CBD = 10^\circ$.

Would you like more details or further clarifications?

Here are 5 related questions to expand your understanding:

1. What other examples can you find for angles that sum to 180° but aren't all acute?
2. How can you determine whether supplementary angles are acute, right, or obtuse?
3. If the area of a rectangle is known, how do you find all possible pairs of length and width?
4. How can you geometrically divide an angle into unequal parts?
5. Can supplementary angles both be obtuse? Why or why not?

Tip: When checking conditions related to angles or areas, always consider possible variations beyond the most obvious cases. This helps in finding counterexamples.