Question: User: 6. “If two lines are perpendicular, then they intersect to form right angles.” Identify the following statements as the contrapositive, converse, or inverse of the statement above.
a. If two lines do not intersect to form right angles, then they are not perpendicular.
b. If two lines are not perpendicular, then they do not form right angles.
c. If two lines intersect to form right angles, then they are perpendicular.
7. Find the following using the given conditional statement. If two angles are a linear pair, then they are supplementary.
a. Converse: ________________________________________________
Is the converse true or false? If true, explain how. If false, provide a counterexample. Circle one: True or False Explanation
b. Inverse: _________________________________________________
Is the inverse true or false? If true, explain how. If false, provide a counterexample. Circle one: True or False Explanation
c. Contrapositive: ____________________________________________
Is the contrapositive true or false? If true, explain how. If false, provide a counterexample. Circle one: True or False Explanation
8. Write 3 true statements from the picture below.
A---B---C
Graph shape and math position_hint: bottom-left; a line segment with endpoints A and C, point B between them, with tick marks indicating AB = BC.
1. **Problem 6:** Identify the statements as contrapositive, converse, or inverse of the statement: "If two lines are perpendicular, then they intersect to form right angles."
2. **Original statement:**
$$\text{If two lines are perpendicular, then they intersect to form right angles.}$$
3. **Definitions:**
- Converse: Switch hypothesis and conclusion.
- Inverse: Negate both hypothesis and conclusion.
- Contrapositive: Switch and negate both hypothesis and conclusion.
4. **Analyze each statement:**
a. "If two lines do not intersect to form right angles, then they are not perpendicular."
- This negates both parts and switches them.
- This is the **contrapositive**.
b. "If two lines are not perpendicular, then they do not form right angles."
- Negates both parts but does not switch.
- This is the **inverse**.
c. "If two lines intersect to form right angles, then they are perpendicular."
- Switches hypothesis and conclusion without negation.
- This is the **converse**.
5. **Problem 7:** Given conditional: "If two angles are a linear pair, then they are supplementary."
6. **Converse:**
$$\text{If two angles are supplementary, then they are a linear pair.}$$
- **Is the converse true or false?**
- **False.** Explanation: Two angles can be supplementary without being adjacent or forming a linear pair (e.g., angles on different parts of a figure).
7. **Inverse:**
$$\text{If two angles are not a linear pair, then they are not supplementary.}$$
- **Is the inverse true or false?**
- **False.** Explanation: Angles can be supplementary without being a linear pair.
8. **Contrapositive:**
$$\text{If two angles are not supplementary, then they are not a linear pair.}$$
- **Is the contrapositive true or false?**
- **True.** Explanation: Since the original statement is true, its contrapositive is also true.
9. **Problem 8:** Given line segment with points A, B, C where B is between A and C and $$AB = BC$$.
10. **Three true statements:**
- $$AB = BC$$ (given by tick marks)
- $$AC = AB + BC$$ (segment addition postulate)
- Point B is the midpoint of $$AC$$ (definition of midpoint since $$AB = BC$$)