# Lesson 8

The Perpendicular Bisector Theorem

### Lesson Narrative

In an earlier unit, students conjectured that they could construct a perpendicular bisector by finding points that were an equal distance from the two endpoints of the segment being bisected. In this lesson, they prove the Perpendicular Bisector Theorem, confirming their conjecture about equidistant points and perpendicular bisectors. The purpose of completing this proof now is to prepare to use the Perpendicular Bisector Theorem in the proof of the Side-Side-Side Triangle Congruence Theorem.

In this lesson, students write proofs of both directions of the Perpendicular Bisector Theorem. They look at sample student work, critique it, write their own proofs, and finally critique their partner’s proof (MP3). Syntheses focus on applying the Perpendicular Bisector Theorem and on what it means for two logical statements to be converses of one another. Two statements are converses if the “if” part and the “then” part are swapped.

Technology isn‘t required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.

### Learning Goals

Teacher Facing

• Critique others' reasoning (in written language) about the Perpendicular Bisector Theorem.
• Prove (orally and in writing) the Perpendicular Bisector Theorem.

### Student Facing

• Let’s convince ourselves that what we’ve conjectured about perpendicular bisectors must be true.

### Required Preparation

The student diagrams from a previous lesson‘s cool-down, Revisiting Perpendicular Bisectors, will be used in the Not Too Close, Not Too Far activity.

Prepare additional copies of the Blank Reference Chart blackline master (double sided, 1 per student). Students can staple the new chart to their full ones, as they will need to continue to refer to the whole packet.

### Student Facing

• I can critique an explanation of the Perpendicular Bisector Theorem.
• I can explain why the Perpendicular Bisector Theorem is true.

Building On

Building Towards

### Glossary Entries

• auxiliary line

An extra line drawn in a figure to reveal hidden structure.

For example, the line shown in the isosceles triangle is a line of symmetry, and the lines shown in the parallelogram suggest a way of rearranging it into a rectangle.

• converse

The converse of an if-then statement is the statement that interchanges the hypothesis and the conclusion. For example, the converse of "if it's Tuesday, then this must be Belgium" is "if this is Belgium, then it must be Tuesday."

• corresponding

For a rigid transformation that takes one figure onto another, a part of the first figure and its image in the second figure are called corresponding parts. We also talk about corresponding parts when we are trying to prove two figures are congruent and set up a correspondence between the parts to see if the parts are congruent.

In the figure, segment $$AB$$ corresponds to segment $$DE$$, and angle $$BCA$$ corresponds to angle $$EFD$$.

• parallelogram

A quadrilateral in which pairs of opposite sides are parallel.