https://file.aiquickdraw.com/mathbot/file/1727657094167nrepn8n7.jpg

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.

Show that each statement is false by providing a counterexample.

Explore how to disprove mathematical statements related to rectangles, complementary angles, line segments, and triangle angles by providing counterexamples. Perfect for students in Grades 7-9 learning about geometry and proofs.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation