Lesson 4 It’s All in Your Head Solidify Understanding

Jump Start

1.

Given , do the following:

  1. Construct an equilateral triangle using as one of the sides.

  2. Construct the perpendicular bisector of .

  3. Construct an isosceles triangle using as the base.

Line segment AB

2.

Can you construct an isosceles triangle with the base whose vertex would not lie on the perpendicular bisector of ? Why or why not?

Line segment AB

3.

How does this work relate to the Perpendicular Bisector Theorem we proved in the previous task?

Learning Focus

Select and sequence statements for a proof using flow diagrams.

Define lines and line segments related to triangles: medians, altitudes, angle bisectors, and perpendicular bisectors of the sides.

When a lot of things are true about a diagram, how can we identify and organize the statements that must be used to justify a particular claim?

Besides the line segments that form the sides, what other lines and line segments are useful in describing features of a triangle?

Open Up the Math: Launch, Explore, Discuss

In the previous task you were asked to justify some claims by writing paragraphs explaining how various figures were constructed and how those constructions convinced you that the claims were true. Perhaps you found it difficult to say everything you felt you just knew. Sometimes we all find it difficult to explain our ideas and to get those ideas out of our heads and written down on paper.

Organizing ideas and breaking complex relationships down into smaller chunks can make the task of proving a claim more manageable. One way to do this is to use a flow diagram.

1.

Which One Doesn’t Belong?

Read each definition and examine the accompanying diagram. Think of a reason why each definition might not belong with the other three.

A.
Triangle ABC with altitude AD.

In a triangle, an altitude is a line segment drawn from a vertex perpendicular to the opposite side (or an extension of the opposite side).

B.
Triangle ABC with median AE.

In a triangle, a median is a line segment drawn from a vertex to the midpoint of the opposite side.

C.
Triangle ABC with angle bisector BX.

In a triangle, an angle bisector is a line segment or ray drawn from a vertex that cuts the angle in half.

D.
Triangle ABC with perpendicular bisector from the midpoint of AC to point P on AB.

In a triangle, a perpendicular bisector of a side is a line drawn perpendicular to a side of the triangle through its midpoint.

Reason:

Travis used a compass and straightedge to construct an equilateral triangle. He then folded his diagram across the two points of intersection of the circles to construct a line of reflection. Travis, Tehani, Carlos, and Clarita are trying to decide what to name the line segment from to .

  • Travis thinks the line segment they have constructed is also a median of the equilateral triangle.

  • Tehani thinks it is an angle bisector.

  • Clarita thinks it is an altitude.

  • Carlos thinks it is a perpendicular bisector of the opposite side.

A circle with Center A and a circle with Center B, Circle A and B overlap such that they share a radius that forms a side of a triangle AB and connects with a point C on both circles creating Triangle ABC.

The four friends are trying to convince each other that they are right.

A flow diagram shows the statements that can be written to describe relationships in the diagram, or conclusions that can be made by connecting multiple ideas. You will use the flow diagram to identify the statements each of the students—Travis, Tehani, Carlos, and Clarita—might use to make their case. Answer the following problems about what each student needs to know about the line of reflection to support their claim.

2.

To support his claim that the line of reflection is a median of the equilateral triangle, Travis will need to show that:

3.

To support her claim that the line of reflection is an angle bisector of the equilateral triangle, Tehani will need to show that:

4.

To support her claim that the line of reflection is an altitude of the equilateral triangle, Clarita will need to show that:

5.

To support his claim that the line of reflection is a perpendicular bisector of a side of the equilateral triangle, Carlos will need to show that:

Provided is a flow diagram of statements that can be written to describe relationships in the diagram or conclusions that can be made by connecting multiple ideas.

A circle with Center A and a circle with Center B, Circle A and B overlap such that they share a radius that forms a side of a triangle AB and connects with a point C on both circles creating Triangle ABC.

6.

Use four different colors to identify the statements each of the students—Travis, Tehani, Clarita, and Carlos might use to make their case.

a graphic representing what the students statements are

7.

Label each of the arrows and braces in the flow diagram with one of the following reasons that justifies why you can make the connection between the statement (or statements) previously accepted as true and the conclusion that follows:

1. Definition of reflection

2. Definition of translation

3. Definition of rotation

4. Definition of an equilateral triangle

5. Definition of perpendicular

6. Definition of midpoint

7. Definition of altitude

8. Definition of median

9. Definition of angle bisector

10. Definition of perpendicular bisector

11. Equilateral triangles can be folded onto themselves about a line of reflection

12. Equilateral triangles can be rotated onto themselves

13. SSS triangle congruence criteria

14. SAS triangle congruence criteria

15. ASA triangle congruence criteria

16. Corresponding parts of congruent triangles are congruent

17. Reflexive Property

a graphic representing what the students statements are

Pause and Reflect

Travis and his friends have seen their teacher write two-column proofs in which the reasons justifying a statement are written next to the statement being made. Travis decides to turn his argument into a two-column proof, as follows.

Statements

Reasons

is equilateral

Given

is a line of reflection

Equilateral triangles can be folded onto themselves about a line of reflection

is the midpoint of

Definition of reflection

Definition of midpoint

is a median

Definition of median

8.

Write each of Clarita’s, Tehani’s, and Carlos’s arguments in two-column proof format.

a.

Clarita’s proof:

b.

Carlos’ proof:

c.

Tehani’s proof:

Ready for More?

Create a flow diagram of all of the geometric ideas and relationships contained in the following diagram, then select one of the following conjectures and create a flow proof of the selected conjecture that includes only the necessary information from the flow diagram needed to prove the conjecture.

Conjecture: The quadrilateral is a rhombus.

Conjecture: The diagonals bisect each other.

Conjecture: The diagonals are perpendicular.

Conjecture: The diagonals bisect the vertex angles.

A circle with Center A and a circle with Center B, Circle A and B overlap such that they share a radius that forms a side of a triangle AB and connects with a point C on both circles creating Triangle ABC. Another triangle is created using AB and a point on both circles, point D, creating Triangle ABD. A line segment is drawn from D to C. Line segment CD intersects AB at point P.

Takeaways

I can generate a “proof idea” using a flow diagram by:

To finalize my flow diagram proof, I should:

Adding Notation, Vocabulary, and Conventions

In the previous task we proved this statement: If a point is on the perpendicular bisector of a line segment, it is equidistant from the endpoints of the segment.

We used this theorem in the Jump Start when we constructed by selecting the 3rd vertex of the triangle to be a point on the perpendicular bisector of .

Statements of the form, “If statement is true, then statement is true,” are called conditional statements.

The converse of a conditional statement is of the form

Write the converse of the theorem we proved about points on the perpendicular bisector in the previous task:

If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

We used this theorem in the jump start when we constructed by selecting the 3rd vertex of the triangle to be the point of intersection of the two circles .

Lesson Summary

In this lesson, we examined a new tool for identifying all of the relationships that exist among the parts of a geometric figure—the flow diagram. Once these relationships have been identified, we can select those that are necessary to prove a particular claim. Therefore, the flow diagram is a tool for selecting and sequencing statements that can be used to create a logical argument for proving new theorems.

Retrieval

Identify each quadrilateral as a parallelogram, rectangle, rhombus, square, or none of these.

1.

Quadrilateral with opposite sides marked congruent.

A.

parallelogram

B.

rectangle

C.

rhombus

D.

square

E.

none of these

2.

Quadrilateral with four right angles.

A.

parallelogram

B.

rectangle

C.

rhombus

D.

square

E.

none of these

3.

Quadrilateral with all sides marked congruent.

A.

parallelogram

B.

rectangle

C.

rhombus

D.

square

E.

none of these

4.

After a translation occurs and the corresponding points on the image and pre-image are connected to form line segments, what will be true of these newly formed line segments between corresponding points? Draw a sketch to illustrate your response.