3. If two angles are vertical angles then they are adjacent. Never 4. If two angles are supplementary then one angle is acute and one angle is obtuse. Use the figure at the right. Given one angle measure, find the other three. 5. If $$ m \angle 6=38^{\circ} \rightarrow $$ 6. If $$ m \angle 8=84^{\circ} \rightarrow $$ 7. If $$ m \angle 9=136^{\circ} \rightarrow $$ 8. If $$ m \angle 7=27^{\circ} \rightarrow $$ Exercises $$ 9-12 $$, assume $$ \angle A $$ and $$ \angle B $$ are complementary and $$ \angle B $$ $$ \angle C $$ are supplementary. If $$ m \angle A=52^{\circ} $$, then $$ m \angle B=38 $$ ? $$ \quad $$ and $$ m \angle C-14290-52 \rightarrow 38^{\circ} $$ If $$ m \angle R-

Question

3. If two angles are vertical angles then they are adjacent. Never 4. If two angles are supplementary then one angle is acute and one angle is obtuse. Use the figure at the right. Given one angle measure, find the other three. 5. If $$ m \angle 6=38^{\circ} \rightarrow $$ 6. If $$ m \angle 8=84^{\circ} \rightarrow $$ 7. If $$ m \angle 9=136^{\circ} \rightarrow $$ 8. If $$ m \angle 7=27^{\circ} \rightarrow $$ Exercises $$ 9-12 $$, assume $$ \angle A $$ and $$ \angle B $$ are complementary and $$ \angle B $$ $$ \angle C $$ are supplementary. If $$ m \angle A=52^{\circ} $$, then $$ m \angle B=38 $$ ? $$ \quad $$ and $$ m \angle C-14290-52 \rightarrow 38^{\circ} $$ If $$ m \angle R-<\rightarrow 0 $$

UpStudy AI · Accepted Answer

Sure, let's go through each of your problems step by step. Since some information might be missing (like the specific figure referenced), I'll make reasonable assumptions based on standard geometric principles regarding vertical, supplementary, and complementary angles. If you have a specific figure or additional details, feel free to provide them for more tailored assistance.

---

### **Problem 3:**
**Statement:** *If two angles are vertical angles then they are adjacent.*

**Answer:**  
**False.**  
**Explanation:** Vertical angles are formed when two lines intersect, creating opposite pairs of equal angles. They **are not** adjacent because they do not share a common side. Adjacent angles share a common vertex and a common side, but vertical angles are across from each other.

---

### **Problem 4:**
**Statement:** *If two angles are supplementary then one angle is acute and one angle is obtuse.*

**Answer:**  
**False.**  
**Explanation:**  
While it's true that one acute angle (less than 90°) and one obtuse angle (more than 90°) can be supplementary (summing to 180°), it's **not the only possibility**. Both angles can also be right angles (each measuring exactly 90°), which are neither acute nor obtuse. Therefore, the statement is not always true.

---

### **Problems 5-8:**
**Instructions:** *Use the figure at the right. Given one angle measure, find the other three.*

*Without the specific figure, we'll assume the angles are related as vertical and supplementary angles. Let's denote the angles as follows for clarity:*

- **Angle 6 (∠6)**
- **Angle 7 (∠7)**
- **Angle 8 (∠8)**
- **Angle 9 (∠9)**

**General Relationships:**
- **Vertical Angles:** Opposite angles formed by two intersecting lines. They are equal in measure.
- **Supplementary Angles:** Two angles whose measures add up to 180°.

---

#### **Problem 5: If $$ m\angle 6 = 38^\circ $$, find the measures of Angles 7, 8, and 9.**

**Solution:**
1. **Vertical Angle:**
   - $$ \angle 6 $$ is vertical to $$ \angle 8 $$.
   - Therefore, $$ m\angle 8 = m\angle 6 = 38^\circ $$.

2. **Supplementary Angles:**
   - $$ \angle 6 $$ and $$ \angle 7 $$ are supplementary.
   - $$ m\angle 7 = 180^\circ - 38^\circ = 142^\circ $$.

3. **Vertical Angle:**
   - $$ \angle 7 $$ is vertical to $$ \angle 9 $$.
   - Therefore, $$ m\angle 9 = m\angle 7 = 142^\circ $$.

**Summary:**
- $$ m\angle 6 = 38^\circ $$
- $$ m\angle 8 = 38^\circ $$ (Vertical to ∠6)
- $$ m\angle 7 = 142^\circ $$ (Supplementary to ∠6)
- $$ m\angle 9 = 142^\circ $$ (Vertical to ∠7)

---

#### **Problem 6: If $$ m\angle 8 = 84^\circ $$, find the measures of Angles 6, 7, and 9.**

**Solution:**
1. **Vertical Angle:**
   - $$ \angle 8 $$ is vertical to $$ \angle 6 $$.
   - Therefore, $$ m\angle 6 = m\angle 8 = 84^\circ $$.

2. **Supplementary Angles:**
   - $$ \angle 6 $$ and $$ \angle 7 $$ are supplementary.
   - $$ m\angle 7 = 180^\circ - 84^\circ = 96^\circ $$.

3. **Vertical Angle:**
   - $$ \angle 7 $$ is vertical to $$ \angle 9 $$.
   - Therefore, $$ m\angle 9 = m\angle 7 = 96^\circ $$.

**Summary:**
- $$ m\angle 8 = 84^\circ $$
- $$ m\angle 6 = 84^\circ $$ (Vertical to ∠8)
- $$ m\angle 7 = 96^\circ $$ (Supplementary to ∠6)
- $$ m\angle 9 = 96^\circ $$ (Vertical to ∠7)

---

#### **Problem 7: If $$ m\angle 9 = 136^\circ $$, find the measures of Angles 6, 7, and 8.**

**Solution:**
1. **Vertical Angle:**
   - $$ \angle 9 $$ is vertical to $$ \angle 7 $$.
   - Therefore, $$ m\angle 7 = m\angle 9 = 136^\circ $$.

2. **Supplementary Angles:**
   - $$ \angle 7 $$ and $$ \angle 6 $$ are supplementary.
   - $$ m\angle 6 = 180^\circ - 136^\circ = 44^\circ $$.

3. **Vertical Angle:**
   - $$ \angle 6 $$ is vertical to $$ \angle 8 $$.
   - Therefore, $$ m\angle 8 = m\angle 6 = 44^\circ $$.

**Summary:**
- $$ m\angle 9 = 136^\circ $$
- $$ m\angle 7 = 136^\circ $$ (Vertical to ∠9)
- $$ m\angle 6 = 44^\circ $$ (Supplementary to ∠7)
- $$ m\angle 8 = 44^\circ $$ (Vertical to ∠6)

---

#### **Problem 8: If $$ m\angle 7 = 27^\circ $$, find the measures of Angles 6, 8, and 9.**

**Solution:**
1. **Vertical Angle:**
   - $$ \angle 7 $$ is vertical to $$ \angle 9 $$.
   - Therefore, $$ m\angle 9 = m\angle 7 = 27^\circ $$.

2. **Supplementary Angles:**
   - $$ \angle 7 $$ and $$ \angle 6 $$ are supplementary.
   - $$ m\angle 6 = 180^\circ - 27^\circ = 153^\circ $$.

3. **Vertical Angle:**
   - $$ \angle 6 $$ is vertical to $$ \angle 8 $$.
   - Therefore, $$ m\angle 8 = m\angle 6 = 153^\circ $$.

**Summary:**
- $$ m\angle 7 = 27^\circ $$
- $$ m\angle 9 = 27^\circ $$ (Vertical to ∠7)
- $$ m\angle 6 = 153^\circ $$ (Supplementary to ∠7)
- $$ m\angle 8 = 153^\circ $$ (Vertical to ∠6)

---

### **Exercises 9-12:**

**Instructions:** *Assume $$ \angle A $$ and $$ \angle B $$ are complementary, and $$ \angle B $$ and $$ \angle C $$ are supplementary. If $$ m\angle A = 52^\circ $$, find $$ m\angle B $$ and $$ m\angle C $$.*

**Definitions:**
- **Complementary Angles:** Two angles whose measures add up to 90°.
- **Supplementary Angles:** Two angles whose measures add up to 180°.

**Given:**
- $$ m\angle A = 52^\circ $$
- $$ \angle A $$ and $$ \angle B $$ are complementary.
- $$ \angle B $$ and $$ \angle C $$ are supplementary.

**Find:**
- $$ m\angle B $$
- $$ m\angle C $$

**Solution:**

1. **Find $$ m\angle B $$ (Complementary to $$ \angle A $$):**
   $$
   m\angle A + m\angle B = 90^\circ \
   52^\circ + m\angle B = 90^\circ \
   m\angle B = 90^\circ - 52^\circ = 38^\circ
   $$

2. **Find $$ m\angle C $$ (Supplementary to $$ \angle B $$):**
   $$
   m\angle B + m\angle C = 180^\circ \
   38^\circ + m\angle C = 180^\circ \
   m\angle C = 180^\circ - 38^\circ = 142^\circ
   $$

**Summary:**
- $$ m\angle A = 52^\circ $$
- $$ m\angle B = 38^\circ $$ (Complementary to ∠A)
- $$ m\angle C = 142^\circ $$ (Supplementary to ∠B)

---

### **Additional Note:**
**Statement:** *If $$ m\angle R $$ → 0°*

**Interpretation & Explanation:**

The statement seems incomplete or possibly a typographical error. However, if it's intended to address what happens when an angle ($$ \angle R $$) approaches 0°, here's an interpretation:

- **As $$ m\angle R $$ approaches 0°,** the two sides of the angle become closer to being a straight line, effectively making the angle negligible. This can be relevant in various geometric contexts, such as limits, when discussing the behavior of shapes, or in trigonometric approximations.

If you have more context or a complete statement regarding $$ \angle R $$, please provide it for a more accurate explanation.

---

Feel free to ask if you need further clarification on any of these problems or if there's more information you'd like to explore!
**Summary of Solutions:** 1. **Problem 3:** *False.* Vertical angles are not adjacent. 2. **Problem 4:** *False.* Two supplementary angles can both be right angles (each 90°). 3. **Problems 5-8:** - **Problem 5:** - $$ m\angle 6 = 38^\circ $$ - $$ m\angle 8 = 38^\circ $$ - $$ m\angle 7 = 142^\circ $$ - $$ m\angle 9 = 142^\circ $$ - **Problem 6:** - $$ m\angle 8 = 84^\circ $$ - $$ m\angle 6 = 84^\circ $$ - $$ m\angle 7 = 96^\circ $$ - $$ m\angle 9 = 96^\circ $$ - **Problem 7:** - $$ m\angle 9 = 136^\circ $$ - $$ m\angle 7 = 136^\circ $$ - $$ m\angle 6 = 44^\circ $$ - $$ m\angle 8 = 44^\circ $$ - **Problem 8:** - $$ m\angle 7 = 27^\circ $$ - $$ m\angle 9 = 27^\circ $$ - $$ m\angle 6 = 153^\circ $$ - $$ m\angle 8 = 153^\circ $$ 4. **Exercises 9-12:** - Given $$ m\angle A = 52^\circ $$, - $$ m\angle B = 38^\circ $$ (Complementary to ∠A) - $$ m\angle C = 142^\circ $$ (Supplementary to ∠B) 5. **Additional Note:** *If $$ m\angle R $$ approaches 0°, the angle becomes negligible, with its sides nearly aligned.* **Conclusion:** The provided problems cover various aspects of angle relationships, including vertical, supplementary, and complementary angles. The solutions demonstrate how to find missing angle measures based on these relationships.

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