Let's go through each of the statements and find counterexamples to show that they are false.

(a) If the area of a rectangle is 100, then the length is 25 and the width is 4.

Counterexample:
The area of a rectangle is given by $\text{Area} = \text{length} \times \text{width}$. There are many pairs of values for length and width that result in an area of 100. For example:

• Length = 10, Width = 10.

Thus, the given statement is false because the length doesn't have to be 25 and the width doesn't have to be 4 for the area to be 100.

(b) If $\angle 1$ and $\angle 2$ are complementary angles, then one of them must have a measure less than 45°.

Counterexample:
Complementary angles sum to 90°. If $\angle 1 = 45^\circ$ and $\angle 2 = 45^\circ$, both angles are equal to 45° and neither is less than 45°. Therefore, the statement is false.

(c) If the length of $\overline{AC}$ is 46 and point B lies on $\overline{AC}$, then $AB = 23$ and $BC = 23$.

Counterexample:
For a point B on $\overline{AC}$, it does not necessarily have to divide the segment into two equal parts. For example, we can have:

• $AB = 10$
• $BC = 36$

This would still make the total length of $\overline{AC} = 46$, but the division is unequal. Therefore, the statement is false.

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

Counterexample:
If all three angles are less than 90° but still sum to 180°, then none of them are obtuse. For example:

• $m \angle R = 60^\circ$
• $m \angle S = 60^\circ$
• $m \angle T = 60^\circ$

None of these angles are obtuse, yet their sum is still 180°. Thus, the statement is false.

Would you like further details or clarifications?

Here are some related questions for deeper understanding:

1. What is the relationship between length, width, and area of a rectangle?
2. Can two complementary angles both be greater than 45°?
3. What are some other ways to divide a segment that don’t result in equal parts?
4. What conditions determine whether an angle is acute, right, or obtuse?
5. How do you determine if a triangle has no obtuse angles?

Tip: Always remember to test edge cases or extreme values when checking the validity of a mathematical statement.