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

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.