\begin{tabular}{l|l|}\hline $$ \begin{array}{l} ext { Sample Response: Because vertical angles are } \ ext { congruent, angles } 1 ext { and } 4 ext { are congruent. Because } \ ext { corresponding angles are congruent, angles } 4 ext { and } 8 ext { are } \ \begin{array}{l} ext { congruent. Because angle } 4 ext { is congruent to both } 1 ext { and } 8, \ ext { angle } 1 ext { is congruent to angle } 8 .\end{array}\end{array} $$ & $$ \begin{array}{l} ext { What did you include in your response? Check all that } \ ext { apply. } \ \square ext { vertical angles theorem } \ \square ext { corresponding angles theorem } \ \square ext { transitive property }\end{array} $$ \ $$ \square $$ supplementary angles \ $$ \square $$ linear pairs \end{tabular}

Question

\begin{tabular}{l|l|}\hline $$ \begin{array}{l}	ext { Sample Response: Because vertical angles are } \ 	ext { congruent, angles } 1 	ext { and } 4 	ext { are congruent. Because } \ 	ext { corresponding angles are congruent, angles } 4 	ext { and } 8 	ext { are } \ \begin{array}{l}	ext { congruent. Because angle } 4 	ext { is congruent to both } 1 	ext { and } 8, \ 	ext { angle } 1 	ext { is congruent to angle } 8 .\end{array}\end{array} $$ & $$ \begin{array}{l}	ext { What did you include in your response? Check all that } \ 	ext { apply. } \ \square 	ext { vertical angles theorem } \ \square 	ext { corresponding angles theorem } \ \square 	ext { transitive property }\end{array} $$ \ $$ \square $$ supplementary angles \ $$ \square $$ linear pairs \end{tabular}

UpStudy AI · Accepted Answer

To analyze the provided sample response and check the included concepts, let's break down the reasoning step by step.

1. **Vertical Angles Theorem**: The response states that angles 1 and 4 are congruent because they are vertical angles. This is a direct application of the vertical angles theorem, which states that vertical angles are always congruent.

2. **Corresponding Angles Theorem**: The response then states that angles 4 and 8 are congruent because they are corresponding angles. This is an application of the corresponding angles theorem, which states that when two parallel lines are cut by a transversal, the corresponding angles are congruent.

3. **Transitive Property**: The response concludes that angle 1 is congruent to angle 8 because angle 4 is congruent to both angle 1 and angle 8. This uses the transitive property of equality, which states that if $$ a = b $$ and $$ b = c $$, then $$ a = c $$.

Now, let's check the options provided in the second column:

- **Vertical Angles Theorem**: ✔️ (Included)
- **Corresponding Angles Theorem**: ✔️ (Included)
- **Transitive Property**: ✔️ (Included)
- **Supplementary Angles**: ❌ (Not included)
- **Linear Pairs**: ❌ (Not included)

### Summary of Included Concepts:
- Vertical Angles Theorem: Yes
- Corresponding Angles Theorem: Yes
- Transitive Property: Yes
- Supplementary Angles: No
- Linear Pairs: No

Thus, the response effectively includes the vertical angles theorem, the corresponding angles theorem, and the transitive property, while it does not include supplementary angles or linear pairs.
The response includes the vertical angles theorem, the corresponding angles theorem, and the transitive property. Supplementary angles and linear pairs are not mentioned.

supplementary angles

linear pairs

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