Logical Reasoning Mixed Practice worksheet for students to practice identifying and transforming conditional statements.
Logical Reasoning Mixed Practice worksheet with statements to identify as conditional statements, underline hypotheses, write converses, inverses, and contrapositives, and provide a counterexample.
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Step-by-step solution for: Geometry Worksheet: Mixed Practice - Logical Reasoning
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Show Answer Key & Explanations
Step-by-step solution for: Geometry Worksheet: Mixed Practice - Logical Reasoning
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> A conditional statement is in the form:
> If [hypothesis], then [conclusion].
> The *hypothesis* is the "if" part (the condition).
> The *conclusion* is the "then" part (the result).
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#### 1. Two angles that sum to 180° are supplementary.
✔ Conditional form:
If two angles sum to 180°, then they are supplementary.
- Hypothesis (underline once): two angles sum to 180°
- Conclusion (underline twice): they are supplementary
> ✔ *Explanation*: Supplementary angles are defined as two angles whose measures add up to 180° — so this is a true statement.
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#### 2. A figure is a square, so it is a rectangle.
✔ Conditional form:
If a figure is a square, then it is a rectangle.
- Hypothesis (underline once): a figure is a square
- Conclusion (underline twice): it is a rectangle
> ✔ *Explanation*: All squares are rectangles (because they have four right angles and opposite sides equal), but not all rectangles are squares. So this is logically valid.
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#### 3. The midpoint M of a segment bisects a segment.
✔ Conditional form:
If M is the midpoint of a segment, then M bisects the segment.
- Hypothesis (underline once): M is the midpoint of a segment
- Conclusion (underline twice): M bisects the segment
> ✔ *Explanation*: By definition, the midpoint divides a segment into two equal parts — so it bisects it. True statement.
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#### 4. All “A’s” get you on the “A” honor roll.
✔ Conditional form:
If you get all A’s, then you are on the “A” honor roll.
- Hypothesis (underline once): you get all A’s
- Conclusion (underline twice): you are on the “A” honor roll
> ✔ *Explanation*: This is a school policy statement — assuming the rule is strictly followed, this is conditionally true.
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> Given conditional:
> If an angle has a measure of 100°, then it is obtuse.
First, recall:
- Converse: Switch hypothesis and conclusion.
→ *If it is obtuse, then an angle has a measure of 100°.*
- Inverse: Negate both hypothesis and conclusion.
→ *If an angle does not have a measure of 100°, then it is not obtuse.*
- Contrapositive: Switch AND negate both.
→ *If an angle is not obtuse, then it does not have a measure of 100°.*
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#### 5.
✔ Converse:
If an angle is obtuse, then it has a measure of 100°.
> ❗️*This is FALSE.* An obtuse angle is any angle between 90° and 180° — so it could be 110°, 120°, etc., not necessarily 100°.
✔ Inverse:
If an angle does not have a measure of 100°, then it is not obtuse.
> ❗️*This is FALSE.* For example, an angle of 110° is not 100°, but it’s still obtuse.
✔ Contrapositive:
If an angle is not obtuse, then it does not have a measure of 100°.
> ✔ *This is TRUE.* Because if an angle is not obtuse, it must be ≤90° or ≥180° — so it cannot be 100°.
> *(Note: In logic, the contrapositive of a true conditional is always true.)*
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✔ Counterexample:
(-3) × (4) = -12
> ✔ *Explanation*: Robert’s conjecture is false because multiplying a negative number and a positive number gives a negative result.
> Another counterexample: (5) × (-2) = -10
> 📝 *Note:* The product is only positive if both numbers are positive OR both are negative. If one is positive and one is negative, the product is negative.
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## ✔ Final Answers Summary:
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1. If two angles sum to 180°, then they are supplementary.
→ Hypothesis: _two angles sum to 180°_
→ Conclusion: __they are supplementary__
2. If a figure is a square, then it is a rectangle.
→ Hypothesis: _a figure is a square_
→ Conclusion: __it is a rectangle__
3. If M is the midpoint of a segment, then M bisects the segment.
→ Hypothesis: _M is the midpoint of a segment_
→ Conclusion: __M bisects the segment__
4. If you get all A’s, then you are on the “A” honor roll.
→ Hypothesis: _you get all A’s_
→ Conclusion: __you are on the “A” honor roll__
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5.
- Converse: If an angle is obtuse, then it has a measure of 100°.
- Inverse: If an angle does not have a measure of 100°, then it is not obtuse.
- Contrapositive: If an angle is not obtuse, then it does not have a measure of 100°.
6. Counterexample: (-3) × 4 = -12
*(Any multiplication of a positive and negative number works.)*
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Let me know if you’d like this formatted for printing or turned into a worksheet answer key! 😊
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Part 1: Write the following statements as a conditional statement. Underline the hypothesis once and the conclusion twice.
> A conditional statement is in the form:
> If [hypothesis], then [conclusion].
> The *hypothesis* is the "if" part (the condition).
> The *conclusion* is the "then" part (the result).
---
#### 1. Two angles that sum to 180° are supplementary.
✔ Conditional form:
If two angles sum to 180°, then they are supplementary.
- Hypothesis (underline once): two angles sum to 180°
- Conclusion (underline twice): they are supplementary
> ✔ *Explanation*: Supplementary angles are defined as two angles whose measures add up to 180° — so this is a true statement.
---
#### 2. A figure is a square, so it is a rectangle.
✔ Conditional form:
If a figure is a square, then it is a rectangle.
- Hypothesis (underline once): a figure is a square
- Conclusion (underline twice): it is a rectangle
> ✔ *Explanation*: All squares are rectangles (because they have four right angles and opposite sides equal), but not all rectangles are squares. So this is logically valid.
---
#### 3. The midpoint M of a segment bisects a segment.
✔ Conditional form:
If M is the midpoint of a segment, then M bisects the segment.
- Hypothesis (underline once): M is the midpoint of a segment
- Conclusion (underline twice): M bisects the segment
> ✔ *Explanation*: By definition, the midpoint divides a segment into two equal parts — so it bisects it. True statement.
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#### 4. All “A’s” get you on the “A” honor roll.
✔ Conditional form:
If you get all A’s, then you are on the “A” honor roll.
- Hypothesis (underline once): you get all A’s
- Conclusion (underline twice): you are on the “A” honor roll
> ✔ *Explanation*: This is a school policy statement — assuming the rule is strictly followed, this is conditionally true.
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Part 2: Write the converse, inverse, and contrapositive
> Given conditional:
> If an angle has a measure of 100°, then it is obtuse.
First, recall:
- Converse: Switch hypothesis and conclusion.
→ *If it is obtuse, then an angle has a measure of 100°.*
- Inverse: Negate both hypothesis and conclusion.
→ *If an angle does not have a measure of 100°, then it is not obtuse.*
- Contrapositive: Switch AND negate both.
→ *If an angle is not obtuse, then it does not have a measure of 100°.*
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#### 5.
✔ Converse:
If an angle is obtuse, then it has a measure of 100°.
> ❗️*This is FALSE.* An obtuse angle is any angle between 90° and 180° — so it could be 110°, 120°, etc., not necessarily 100°.
✔ Inverse:
If an angle does not have a measure of 100°, then it is not obtuse.
> ❗️*This is FALSE.* For example, an angle of 110° is not 100°, but it’s still obtuse.
✔ Contrapositive:
If an angle is not obtuse, then it does not have a measure of 100°.
> ✔ *This is TRUE.* Because if an angle is not obtuse, it must be ≤90° or ≥180° — so it cannot be 100°.
> *(Note: In logic, the contrapositive of a true conditional is always true.)*
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6. Robert made the conjecture: “When two numbers are multiplied together, the answer is always positive.” Provide a counterexample.
✔ Counterexample:
(-3) × (4) = -12
> ✔ *Explanation*: Robert’s conjecture is false because multiplying a negative number and a positive number gives a negative result.
> Another counterexample: (5) × (-2) = -10
> 📝 *Note:* The product is only positive if both numbers are positive OR both are negative. If one is positive and one is negative, the product is negative.
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## ✔ Final Answers Summary:
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Part 1:
1. If two angles sum to 180°, then they are supplementary.
→ Hypothesis: _two angles sum to 180°_
→ Conclusion: __they are supplementary__
2. If a figure is a square, then it is a rectangle.
→ Hypothesis: _a figure is a square_
→ Conclusion: __it is a rectangle__
3. If M is the midpoint of a segment, then M bisects the segment.
→ Hypothesis: _M is the midpoint of a segment_
→ Conclusion: __M bisects the segment__
4. If you get all A’s, then you are on the “A” honor roll.
→ Hypothesis: _you get all A’s_
→ Conclusion: __you are on the “A” honor roll__
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Part 2:
5.
- Converse: If an angle is obtuse, then it has a measure of 100°.
- Inverse: If an angle does not have a measure of 100°, then it is not obtuse.
- Contrapositive: If an angle is not obtuse, then it does not have a measure of 100°.
6. Counterexample: (-3) × 4 = -12
*(Any multiplication of a positive and negative number works.)*
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Let me know if you’d like this formatted for printing or turned into a worksheet answer key! 😊
Parent Tip: Review the logic above to help your child master the concept of logic statements worksheet.
