In this activity, students provide reasons for a proof that parallel lines must have equal slopes, as they prepare to construct an argument (MP3). To prepare them for this reasoning, students begin by noticing and wondering about an image of parallel lines on a coordinate plane. 

Monitor for students who approach the proof using these methods, from more general to more precise:

• Describing properties of the diagram and transformations.
• Annotating the diagram with congruent angles and segments.
• Labeling points and specifying congruent angles and segments using the labels.

Note that this reasoning works only for unique, non-vertical parallel lines that have a nonzero slope. It is not necessary that students make this observation at this time, but it may be worth pointing out if students ask.

Invite students to share their reasoning in response to each question. Highlight any students who annotated their image to make their explanations clearer.

Here are some additional questions for discussion:

• “What if the lines are horizontal?” (All horizontal lines are parallel to each other and have a slope of 0, so the idea of parallel lines having equal slopes applies to this situation.)
• “What if the lines are vertical?” (Slopes of vertical lines are undefined, so the equal slopes criterion does not apply to them.)

Tell students that the converse is also true: If two lines have equal slopes, then they are parallel. Add the following theorem to the class reference chart, and ask students to add it to their reference charts:

Lines are parallel if and only if they have equal slopes. (Theorem)

In this activity, students use the result of a previous proof that parallel lines have the same slope to show that the opposite sides of a quadrilateral have the same slope and are therefore parallel. This leads to the conclusion that the quadrilateral is a parallelogram. This gives an opportunity for students to apply the proof and prepares them for proofs about parallelograms in later courses. Students do not need to formally prove that a figure on a grid is a parallelogram at this time, but students may recognize that two pairs of intersecting parallel lines form a parallelogram. Students make use of the structure of translation by a directed line segment to make observations about slope and parallel lines (MP7).

Monitor for students who use different equivalent equations to represent the parallel lines and ask them to share in the whole-class discussion.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Invite students to share their equations for the lines and how they came up with them. Here are some questions to elicit further discussion:

• "What are some different ways to find the equation of the line AC?" 
  • First graph the translation, and then see that it has a slope of 2 and -intercept of 9, so the equation of the line is .
  • Recognize that a translation takes a line to a parallel line, so the slopes are the same. Use the slope of 2 and the new point to find the equation .
  • See that the directed line segment is a shift up 2 and left 2, so it transforms the original line to .
• "How did you know that the 2 lines you drew via the translation were parallel?" (They had the same slope. A translation takes one line to a parallel line.)
• "How did you know the directed line segments were parallel?" (Since they have the same slope, they must be parallel. It is also possible to find the equations of the lines that the directed line segments lie on, and you would see that those slopes are the same.)
• "What does this tell us about Quadrilateral ABCD?" (It's a parallelogram, because it has opposite pairs of parallel lines.)