Lesson 3 Can You Say It with Symbols? Solidify Understanding

Jump Start

Here is one more diagram for you to examine.

Describe the sequence of steps you think were used to construct this diagram beginning with the first figure and ending with the second figure.

Learning Focus

Organize and sequence proof statements using a two-column format.

Examine a claim about points on a perpendicular bisector of a line segment.

How do I keep track of all of the statements that need to be recorded in a proof?

How do I attend to the order of statements in a proof, so that ideas that need to be established first come before statements that require prior information?

Open Up the Math: Launch, Explore, Discuss

Tia and Tehani are doing their math homework together. One of the problems is to prove the following statement.

Prove: The points on the perpendicular bisector of a segment are equidistant from the endpoints of the segment.

Tia and Tehani think this diagram will be helpful to prove this statement, but they know they will need to say more than just describing how to create the diagram. Tia starts by describing the things they know, and Tehani tries to keep a written record by jotting notes down on a piece of paper.

1.

In the table, record in symbolic notation what Tehani may have written to keep track of Tia’s statements. In the examples given, note how Tehani is introducing symbols for the lines and points in the diagram, so she can reference them again without using a lot of words.

Tehani’s Notes	Tia’s Statements
Draw $\overset{―}{A B}$ . Locate its midpoint $M$ , and draw a perpendicular line $ℓ$ through the midpoint.	We need to start with a segment and its perpendicular bisector already drawn.
Pick any point $C$ on line $ℓ$ .	We need to show that any point on the perpendicular bisector is equidistant from the two endpoints, so I can pick any arbitrary point on the perpendicular bisector. Let’s call it $C$ .
Prove:	We need to show that this point is the same distance from the two endpoints.
First prove:	If we knew the two triangles were congruent, we could say that the point on the perpendicular bisector is the same distance from each endpoint. So, what do we know about the two triangles that would let us say that they are congruent?
	We know that both triangles contain a right angle.
	And we know that the perpendicular bisector cuts segment $A B$ into two congruent segments.
	Obviously, the segment from $C$ to the midpoint of segment $A B$ is a side of both triangles.
	So, the triangles are congruent by the SAS triangle congruence criteria.
	Since the triangles are congruent, segments $A C$ and $B C$ are congruent.
Any point $C$ on line $ℓ$ , the perpendicular bisector of $\overset{―}{A B}$ , is equidistant from the endpoints $A$ and $B$ .	And, that proves that point $C$ is equidistant from the two endpoints!

2.

Tehani thinks Tia is brilliant, but she would like the ideas to flow more easily from start to finish. Arrange Tehani’s symbolic notes in a way that someone else could follow the argument and see the connections between ideas.

3.

Would your justification be true regardless of where point $C$ is chosen on the perpendicular bisector? Why?

Pause and Reflect

4.

You can use Tia and Tehani’s theorem about points on the perpendicular bisector of a segment to prove that the base angles of an isosceles triangle are congruent. Write a narrative proof for each of the following statements.

a.

Triangle $A B C$ in Tia and Tehani’s diagram is an isosceles triangle.

b.

The base angles, $∠ A$ and $∠ B$ , of the isosceles triangle in Tia and Tehani’s diagram are congruent.

c.

Tia and Tehani’s diagram can be applied to all isosceles triangles.

Ready for More?

Take one or more of the paragraph proofs you wrote in 6a–6e of the previous lesson, Do You See What I See?, and reformat it as a two-column proof.

Takeaways

Writing the story of a diagram, and the conclusions that can be drawn from it, is central to geometric proofs.

Like any other text, a proof is written by an author and has to make sense to a reader, so it must be thoughtfully constructed.

That is:

Proofs can be written in different formats. So far we have seen:

Today we proved the Perpendicular Bisector Theorem:

We also examined the details of a proof for the base angles of an isosceles triangle:

Adding Notation, Vocabulary, and Conventions

The perpendicular bisector of a segment is

Vocabulary

perpendicular bisector
proof: types—flow, two-column, paragraph
properties of equality
symmetric
two-column proof
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned how to write proofs in a two-column format. This format helps us keep track of the logical organization and sequence of statements in a proof so each statement in the proof can be justified based on the statements that come before it. We proved a statement about the points on a perpendicular bisector of a segment and a statement about the base angles of an isosceles triangle.

Retrieval

Determine which of the triangle congruence criteria would be used to prove the triangles are congruent.