Question

if false, draw a counterexample (with numbers). 9. if two angles are vertical angles, then they are congruent. converse: can a biconditional be written? why? 10. if an angle is obtuse, then it has a measure of 120°. inverse: 11. if two lines are parallel, then they do not intersect. contrapositive: 12. if an angle is not acute, then it is obtuse. inverse:

Explanation:

Step1: Analyze statement 9

The statement "If two angles are vertical angles, then they are congruent" is a well - known theorem in geometry and is True (T).

Step2: Find the converse of statement 9

The converse is "If two angles are congruent, then they are vertical angles". This is False (F). For example, two angles in two different equilateral triangles can be congruent but not vertical angles.

Step3: Determine biconditional for statement 9

A biconditional cannot be written because the converse is false. A biconditional requires both the original statement and its converse to be true.

Step4: Analyze statement 10

The statement "If an angle is obtuse, then it has a measure of 120°" is False (F). An obtuse angle has a measure between 90° and 180°, not just 120°.

Step5: Find the inverse of statement 10

The inverse is "If an angle is not obtuse, then it does not have a measure of 120°". This is True (T) since non - obtuse angles (acute or right) do not have a measure of 120°.

Step6: Analyze statement 11

The statement "If two lines are parallel, then they do not intersect" is True (T) by the definition of parallel lines.

Step7: Find the contrapositive of statement 11

The contrapositive is "If two lines intersect, then they are not parallel". This is True (T) as it is logically equivalent to the original statement.

Step8: Analyze statement 12

The statement "If an angle is not acute, then it is obtuse" is False (F) because a non - acute angle could be a right angle (90°) or a straight angle (180°) not just obtuse.

Step9: Find the inverse of statement 12

The inverse is "If an angle is acute, then it is not obtuse". This is True (T) since an acute angle has a measure less than 90° and an obtuse angle has a measure greater than 90°.

Answer:

1. T; Converse: If two angles are congruent, then they are vertical angles. F; Can a biconditional be written? No. Why? Converse is false.
2. F; Inverse: If an angle is not obtuse, then it does not have a measure of 120°. T
3. T; Contrapositive: If two lines intersect, then they are not parallel. T
4. F; Inverse: If an angle is acute, then it is not obtuse. T