If students are stuck, suggest they shade the two triangles and label the lengths of the legs of the triangles with arbitrary letters so it’s easier to talk about them.

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

After students have shared, ask students these questions about additional slope cases:

• “If lines are horizontal, what can we tell about their slopes?” (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.)
• “If the lines are vertical, what can we tell about their slopes?” (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 2 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 develop a proof that a particular quadrilateral is a parallelogram. First, students write the equation of a line that is parallel to a given line, by using an earlier result that the slopes of parallel lines are the same. Monitor for students who write the equation in different forms.

As students prove that a quadrilateral constructed by using lines with the same slope must be a parallelogram, they are constructing viable arguments (MP3). Monitor for different strategies that students use to prove that the quadrilateral is a parallelogram. Here are some approaches students may take, from more common to less common:

• The quadrilateral is composed of 2 pairs of parallel lines.
• Opposite sides of the quadrilateral are congruent.
• One pair of sides can be considered as a translation by a directed line segment, and the other pair of sides is the same directed line segment.
• The diagonals of the quadrilateral bisect each other.

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

If students struggle with proving that a quadrilateral is a parallelogram, here are some questions to elicit student thinking:

• "What are the features of a parallelogram?"
• "What are some ways we have proved that a quadrilateral is a parallelogram in earlier units?" 
• "How could knowing slope or side lengths help?"

If students aren’t sure how to graph lines, here are some questions to connect to earlier work with lines:

• "How could knowing the slope of the line help?"
• "How could you use a slope triangle and the point to graph a line?"

Invite previously selected students to share their equations. Ask students to justify that all of their answers are correct despite looking different. Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see.
Connect the different responses to the learning goals by asking questions, such as: "What are some different ways to find the equation of the line through ?" 

• First graph the translation, and then see that the line has a slope of 2 and -intercept of 9, so the equation of the line is .
• Because the lines are parallel, the slopes must be the same. Use the slope of 2 and the new point to find the equation .
• See that there is a directed line segment from one line to the other that is a shift up 2 and left 2, so it transforms the original line to .

Next, invite previously selected students to share their proofs that the quadrilateral is a parallelogram. Sequence students in order of most common to least common approaches. Invite students to ask clarifying questions as they listen to each other's arguments.