Lesson 8

This activity prompts students to compare four images. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another. In particular, students will be focused on the characteristics of perpendicular lines.

Arrange students in groups of 2–4. Display the figures for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning why a particular item does not belong, and together, find at least one reason each item doesn‘t belong.

Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology they use, such as perpendicular. Also, press students on unsubstantiated claims.

The proof that if a point \(C\) is the same distance from \(A\) as it is from \(B\), then \(C\) must be on the perpendicular bisector of \(AB\) is challenging. Students may struggle to make sense of what it means to prove that a point must be on a line. This proof, like the proof of the Isosceles Triangle Theorem, requires thinking carefully about how to define an auxiliary line.

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

Arrange students in groups of 2. Use the Notice and Wonder instructional routine with this image to launch the activity.

Display the image for all to see. Ask student to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

\(\overline{AP} \cong \overline{BP}\)

Things students may notice:

• It’s a triangle.
• Segment \(PA\) is congruent to segment \(PB\).
• Angle \(A\) is congruent to angle \(B\) (by the Isosceles Triangle Theorem).
• There’s a dotted line.

Things students may wonder:

• Is the dotted line the line of symmetry?
• Is the dotted line the angle bisector of angle \(APB\)?
• Is the dotted line the perpendicular bisector of segment \(AB\)?

The purpose of this launch is to elicit the idea that special lines through \(P\) when segments \(PA\) and \(PB\) are congruent have many overlapping properties, which will be useful when students draw auxiliary lines later in this activity. While students may notice and wonder many things about these images, wondering about the line through \(P\) is an important focus of the discussion.

If students have access to GeoGebra Geometry from Math Tools, suggest that it might be a helpful tool in this activity.

Remind students who struggle with their critique or edits to make use of the tips from the display, such as asking for 3 statements and 3 reasons, or looking for congruent triangles.

Focus discussion on what students proved and the implications. Display the image from the launch of this activity, and ask what must be true about this image, based on what they just proved. (The dotted line is the perpendicular bisector of \(AB\). The dotted line is the line of reflection of \(AB\).)

\(\overline{AP} \cong \overline{BP}\)

Display this image of the construction of a perpendicular bisector and ask students how what they just proved explains why this construction works. (Both circles have radius \(AB\), so the intersection points of the circles are the same distance from \(A\) and \(B\). Therefore, both intersection points are on the perpendicular bisector of \(AB\).)

Ask students to add part 1 of the Perpendicular Bisector Theorem to their reference charts as you add it to the class reference chart:

If a point \(C\) is the same distance from \(A\) as it is from \(B\), \(C\) must be on the perpendicular bisector of \(AB\). (Theorem)

\(\overline{AC} \cong \overline{BC}\), so \(C\) is on the line through midpoint \(M\) perpendicular to \(\overline{AB}\)

In a previous activity, students had a chance to practice giving feedback on proofs. In this activity, they will put their proof-writing and feedback-giving skills to work as they draft proofs and then give feedback to a partner. Students have already drawn diagrams of this situation in an earlier lesson, so the focus of this activity is writing a proof that is clear and easy to understand (MP3). Students will make revisions to the proofs they write in the cool-down, so focus discussion on common concerns.

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

Arrange students in groups of 2.

Display one of the correct, complete diagrams from the cool-down of the Side-Angle-Side Triangle Congruence lesson. Ask students which statement this image illustrates and why. (If a point is on the perpendicular bisector of a line segment, then that point must be the same distance from each endpoint of the segment.)

Students may think this is the same proof as the previous activity. Ask students what the difference is between this claim and that one. (This one is backwards.) Inform students this is the converse, and the converse of a true statement is not always true, so they will need to write a new proof for this claim. 

Students will make revisions to the proofs they write in the cool-down, so focus discussion on common concerns about the draft proofs. Invite a few students to share their progress.