https://file.aiquickdraw.com/mathbot/file/1727115451474cse7fo41.jpg

Let's go through the counterexamples for each statement provided in the image:

### (a) **If the measures of ∠R, ∠S, and ∠T sum to 180°, then one of the angles must be obtuse.**
The given statement is **false** because three angles can sum to 180° without any of them being obtuse (greater than 90°).

**Counterexample:**
- \( m ∠R = 60° \)
- \( m ∠S = 60° \)
- \( m ∠T = 60° \)

In this case, all angles are acute (less than 90°), yet they sum to 180°, disproving the claim.

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### (b) **If \( m ∠WXZ = 60° \), and point Y is in the interior of ∠WXZ, then \( m ∠WXY = 40° \) and \( m ∠YXZ = 20° \).**
This statement is **false** because the interior angles do not necessarily have to match the given values (40° and 20°).

**Counterexample:**
- \( m ∠WXY = 30° \)
- \( m ∠YXZ = 30° \)

Here, the sum is still 60°, but the division of the angles is different, invalidating the assumption.

---

### (c) **If the area of a rectangle is 64, then the length is 8 and the width is 8.**
This statement is **false** because there are other possible dimensions for a rectangle to have an area of 64.

**Counterexample:**
- Length = 16
- Width = 4

Since \( 16 × 4 = 64 \), this provides a valid counterexample with different dimensions than 8 × 8.

---

### (d) **If ∠1 and ∠2 are supplementary angles, then one of them must be acute.**
This statement is **false** because both angles could be right angles (90°), which are not acute.

**Counterexample:**
- \( m ∠1 = 90° \)
- \( m ∠2 = 90° \)

Both angles are right angles, and while they are supplementary (sum to 180°), neither is acute.

---

Would you like further details or clarification on any of the counterexamples? 

Here are five related questions to deepen your understanding:
1. Can a triangle have all angles greater than 90°?
2. What conditions define supplementary angles?
3. What are other possible dimensions of a rectangle with an area of 64?
4. How can you divide an angle into two parts with different measures?
5. Can an obtuse angle be supplementary to another obtuse angle?

**Tip:** Always remember, supplementary angles must sum to 180°, but there is no restriction on either being acute or obtuse!

Show that each statement is false by providing a counterexample.

Explore counterexamples to geometric and angle statements, covering concepts such as the sum of angles in a triangle, supplementary angles, and the area of rectangles. Learn how to disprove false statements in geometry with examples for Grades 7-9.

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