If `alpha,beta` be straight lines in a plane, then check `R_(1)andR_(2)` for being reflexive, symmetric and transitive `alphaR_(1)beta` if `alphabotbetaandalphaR_(2)beta " if "alpha||beta`.

Answer

To determine the properties of the relations \( R_1 \) and \( R_2 \) defined by the conditions of perpendicularity and parallelism of lines \( \alpha \) and \( \beta \), we will check if each relation is reflexive, symmetric, and transitive.

### Definitions of Relations:
- \( \alpha R_1 \beta \) if \( \alpha \) is perpendicular to \( \beta \).
- \( \alpha R_2 \beta \) if \( \alpha \) is parallel to \( \beta \).

### Step 1: Check for Reflexivity

**For \( R_1 \) (Perpendicularity):**
- A relation is reflexive if every element is related to itself.
- For any line \( \alpha \), it cannot be perpendicular to itself. Therefore, \( \alpha R_1 \alpha \) is false.

**Conclusion for \( R_1 \):** Not reflexive.

**For \( R_2 \) (Parallelism):**
- A line \( \alpha \) is always parallel to itself. Therefore, \( \alpha R_2 \alpha \) is true.

**Conclusion for \( R_2 \):** Reflexive.

### Step 2: Check for Symmetry

**For \( R_1 \) (Perpendicularity):**
- If \( \alpha R_1 \beta \) (i.e., \( \alpha \) is perpendicular to \( \beta \)), then \( \beta \) is also perpendicular to \( \alpha \). Thus, \( \beta R_1 \alpha \) is true.

**Conclusion for \( R_1 \):** Symmetric.

**For \( R_2 \) (Parallelism):**
- If \( \alpha R_2 \beta \) (i.e., \( \alpha \) is parallel to \( \beta \)), then \( \beta \) is also parallel to \( \alpha \). Thus, \( \beta R_2 \alpha \) is true.

**Conclusion for \( R_2 \):** Symmetric.

### Step 3: Check for Transitivity

**For \( R_1 \) (Perpendicularity):**
- Assume \( \alpha R_1 \beta \) and \( \beta R_1 \gamma \) (i.e., \( \alpha \) is perpendicular to \( \beta \) and \( \beta \) is perpendicular to \( \gamma \)).
- This does not imply that \( \alpha \) is perpendicular to \( \gamma \). For example, if \( \alpha \) is vertical, \( \beta \) is horizontal, and \( \gamma \) is vertical, then \( \alpha \) and \( \gamma \) are not perpendicular.

**Conclusion for \( R_1 \):** Not transitive.

**For \( R_2 \) (Parallelism):**
- Assume \( \alpha R_2 \beta \) and \( \beta R_2 \gamma \) (i.e., \( \alpha \) is parallel to \( \beta \) and \( \beta \) is parallel to \( \gamma \)).
- This implies that \( \alpha \) is parallel to \( \gamma \). Thus, \( \alpha R_2 \gamma \) is true.

**Conclusion for \( R_2 \):** Transitive.

### Final Summary:
- **Relation \( R_1 \) (Perpendicularity):**
- Reflexive: No
- Symmetric: Yes
- Transitive: No

- **Relation \( R_2 \) (Parallelism):**
- Reflexive: Yes
- Symmetric: Yes
- Transitive: Yes