Solve your doubts and expand your knowledge with IDNLearn.com's extensive Q&A database. Our experts provide prompt and accurate answers to help you make informed decisions on any topic.
Sagot :
In analyzing the proofs provided by Student A and Student B, let's go through each proof carefully to highlight and correct their mistakes.
### Student A's Proof:
1. Statements:
- [tex]\( m \angle CEA = 8x + 10 \)[/tex] and [tex]\( m \angle AED = 6x + 30 \)[/tex]
- [tex]\( \angle CEA \)[/tex] and [tex]\( \angle AED \)[/tex] are a linear pair
- [tex]\( m \angle CEA + m \angle AED = 180^\circ \)[/tex]
- [tex]\( 8x + 10 + 6x + 30 = 180 \)[/tex]
- [tex]\( 14x + 40 = 180 \)[/tex]
- [tex]\( 14x = 140 \)[/tex]
- [tex]\( x = 10 \)[/tex]
2. Reasons:
- Given
- Definition of linear pair
- Linear pairs are supplementary
- Substitution
- Like terms can be combined
- Addition property of equality
- Division property of equality
After reviewing Student A's proof, we find that each step logically follows from the previous statements and the algebraic manipulations are correctly applied. No mistakes are identified in Student A's proof.
Student A's Mistake: None
### Student B's Proof:
Student B's proof is as follows:
1. We are given that [tex]\( m \angle CEA = (8x + 10)^\circ \)[/tex] and [tex]\( m \angle AED = (6x + 30)^\circ \)[/tex].
2. By the definition of vertical angles, we know that [tex]\( \angle CEA \)[/tex] and [tex]\( \angle AED \)[/tex] are vertical angles.
3. Vertical angles have equal measures, so [tex]\( m \angle CEA = m \angle AED \)[/tex].
4. Substituting [tex]\( m \angle CEA = 8x + 10 \)[/tex] and [tex]\( m \angle AED = 6x + 30 \)[/tex] into [tex]\( m \angle CEA = m \angle AED \)[/tex], we get [tex]\( 8x + 10 = 6x + 30 \)[/tex].
5. Using the subtraction property of equality on the variable terms, this becomes [tex]\( 2x + 10 = 30 \)[/tex].
6. Using the subtraction property of equality on the constants, we get [tex]\( 2x = 20 \)[/tex].
7. By the division property of equality, we prove that [tex]\( x = 10 \)[/tex].
Reviewing Student B's proof:
- The initial assertion that [tex]\( \angle CEA \)[/tex] and [tex]\( \angle AED \)[/tex] are vertical angles is incorrect. Vertical angles are formed by two intersecting lines and are opposite each other, which is not the scenario described here. This misinterpretation leads to an incorrect application of the property that vertical angles are equal.
Student B's Mistake: Step 1 - [tex]\( m \angle CEA \)[/tex] and [tex]\( m \angle AED \)[/tex] are not vertical angles.
### Student A's Proof:
1. Statements:
- [tex]\( m \angle CEA = 8x + 10 \)[/tex] and [tex]\( m \angle AED = 6x + 30 \)[/tex]
- [tex]\( \angle CEA \)[/tex] and [tex]\( \angle AED \)[/tex] are a linear pair
- [tex]\( m \angle CEA + m \angle AED = 180^\circ \)[/tex]
- [tex]\( 8x + 10 + 6x + 30 = 180 \)[/tex]
- [tex]\( 14x + 40 = 180 \)[/tex]
- [tex]\( 14x = 140 \)[/tex]
- [tex]\( x = 10 \)[/tex]
2. Reasons:
- Given
- Definition of linear pair
- Linear pairs are supplementary
- Substitution
- Like terms can be combined
- Addition property of equality
- Division property of equality
After reviewing Student A's proof, we find that each step logically follows from the previous statements and the algebraic manipulations are correctly applied. No mistakes are identified in Student A's proof.
Student A's Mistake: None
### Student B's Proof:
Student B's proof is as follows:
1. We are given that [tex]\( m \angle CEA = (8x + 10)^\circ \)[/tex] and [tex]\( m \angle AED = (6x + 30)^\circ \)[/tex].
2. By the definition of vertical angles, we know that [tex]\( \angle CEA \)[/tex] and [tex]\( \angle AED \)[/tex] are vertical angles.
3. Vertical angles have equal measures, so [tex]\( m \angle CEA = m \angle AED \)[/tex].
4. Substituting [tex]\( m \angle CEA = 8x + 10 \)[/tex] and [tex]\( m \angle AED = 6x + 30 \)[/tex] into [tex]\( m \angle CEA = m \angle AED \)[/tex], we get [tex]\( 8x + 10 = 6x + 30 \)[/tex].
5. Using the subtraction property of equality on the variable terms, this becomes [tex]\( 2x + 10 = 30 \)[/tex].
6. Using the subtraction property of equality on the constants, we get [tex]\( 2x = 20 \)[/tex].
7. By the division property of equality, we prove that [tex]\( x = 10 \)[/tex].
Reviewing Student B's proof:
- The initial assertion that [tex]\( \angle CEA \)[/tex] and [tex]\( \angle AED \)[/tex] are vertical angles is incorrect. Vertical angles are formed by two intersecting lines and are opposite each other, which is not the scenario described here. This misinterpretation leads to an incorrect application of the property that vertical angles are equal.
Student B's Mistake: Step 1 - [tex]\( m \angle CEA \)[/tex] and [tex]\( m \angle AED \)[/tex] are not vertical angles.
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. For dependable and accurate answers, visit IDNLearn.com. Thanks for visiting, and see you next time for more helpful information.
