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Part A

Mr. Avila asked his students to write either a two-column proof or a paragraph proof for the question shown. Student A and Student B each made a mistake in their proof. Identify each student's mistake and the step in which they made it.

Type your response in the box.

\begin{tabular}{|c|c|}
\hline
\multicolumn{2}{|c|}{Student A's Proof} \\
\hline
Statement & Reason \\
\hline
[tex]$m \angle CEA = 8x + 10$[/tex] and [tex]$m \angle AED = 6x + 30$[/tex] & given \\
\hline
[tex]$\angle CEA$[/tex] and [tex]$\angle AED$[/tex] are a linear pair. & definition of linear pair \\
\hline
[tex]$m \angle CEA + m \angle AED = 180^{\circ}$[/tex] & Linear pairs are supplementary. \\
\hline
[tex]$8x + 10 + 6x + 30 = 180$[/tex] & substitution \\
\hline
[tex]$14x + 40 = 180$[/tex] & Like terms can be combined. \\
\hline
[tex]$14x = 140$[/tex] & subtraction property of equality \\
\hline
[tex]$x = 10$[/tex] & division property of equality \\
\hline
\end{tabular}

Student B's Proof

We are given that [tex]$m \angle CEA = (8x + 10)^{\circ}$[/tex] and [tex]$m \angle AED = (6x + 30)^{\circ}$[/tex]. By the definition of vertical angles, we know that [tex]$\angle CEA$[/tex] and [tex]$\angle AED$[/tex] are vertical angles. Vertical angles have equal measures, so [tex]$m \angle CEA = m \angle AED$[/tex]. 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]. Using the subtraction property of equality on the variable terms, this becomes [tex]$2x + 10 = 30$[/tex]. Using the subtraction property of equality on the constants, we get [tex]$2x = 20$[/tex]. By the division property of equality, we prove that [tex]$x = 10$[/tex].



Answer :

GinnyAnswer
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.

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