39. Vertical angles are supplements of each other. 40. If an angle is acute, then its complement is greater than its supplement. 41. If two complementary angles are congruent, then the measure of each angle is $$ 45^{\circ} $$.

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Let's analyze each of the statements you've provided:

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**39. Vertical angles are supplements of each other.**

**False.**

- **Explanation:** Vertical angles are pairs of opposite angles formed when two lines intersect. By definition, vertical angles are **congruent**, meaning they have equal measures. However, being supplementary means that the sum of their measures is $$180^{\circ}$$. Vertical angles do **not** necessarily add up to $$180^{\circ}$$ unless each of them is $$90^{\circ}$$. In general, vertical angles can be any measure as long as they are equal to each other.

- *Example:* If two lines intersect and form vertical angles of $$70^{\circ}$$ each, they are congruent but not supplementary since $$70^{\circ} + 70^{\circ} = 140^{\circ} \neq 180^{\circ}$$.

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**40. If an angle is acute, then its complement is greater than its supplement.**

**False.**

- **Explanation:** 
  - An **acute angle** is an angle that measures less than $$90^{\circ}$$.
  - The **complement** of an angle is what, when added to it, equals $$90^{\circ}$$. So, the complement of an acute angle $$ \theta $$ is $$90^{\circ} - \theta$$.
  - The **supplement** of an angle is what, when added to it, equals $$180^{\circ}$$. So, the supplement of $$ \theta $$ is $$180^{\circ} - \theta$$.

For an acute angle $$ \theta $$:
  $$
  \text{Supplement} - \text{Complement} = (180^{\circ} - \theta) - (90^{\circ} - \theta) = 90^{\circ}
  $$
  This means the supplement is always $$90^{\circ}$$ greater than the complement for any acute angle. Therefore, the complement is **less** than the supplement, not greater.

- *Example:* If $$ \theta = 30^{\circ} $$, then:
    - Complement: $$90^{\circ} - 30^{\circ} = 60^{\circ}$$
    - Supplement: $$180^{\circ} - 30^{\circ} = 150^{\circ}$$
    - Here, $$60^{\circ} < 150^{\circ}$$.

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**41. If two complementary angles are congruent, then the measure of each angle is $$45^{\circ}$$.**

**True.**

- **Explanation:** 
  - If two angles are **complementary**, their measures add up to $$90^{\circ}$$.
  - If they are **congruent**, they have equal measures.
  
  Let each angle be $$ \theta $$. Then:
  $$
  \theta + \theta = 90^{\circ} \implies 2\theta = 90^{\circ} \implies \theta = 45^{\circ}
  $$
  
  Therefore, each angle measures $$45^{\circ}$$.

- *Example:* Two congruent complementary angles must each be $$45^{\circ}$$ because $$45^{\circ} + 45^{\circ} = 90^{\circ}$$.

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**Summary:**
- **Statement 39:** False
- **Statement 40:** False
- **Statement 41:** True
**39. Vertical angles are supplements of each other.** *False.* **40. If an angle is acute, then its complement is greater than its supplement.** *False.* **41. If two complementary angles are congruent, then the measure of each angle is $$45^{\circ}$$.** *True.*

39. Vertical angles are supplements of each other. 40. If an angle is acute, then its complement is greater than its supplement. 41. If two complementary angles are congruent, then the measure of each angle is \( 45^{\circ} \).

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