Lesson 4 It’s All in Your Head Solidify Understanding
Construct an equilateral triangle using
as one of the sides.
Construct the perpendicular bisector of
Construct an isosceles triangle using
as the base.
Can you construct an isosceles triangle with the base
How does this work relate to the Perpendicular Bisector Theorem we proved in the previous task?
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.
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.
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).
In a triangle, a median is a line segment drawn from a vertex to the midpoint of the opposite side.
In a triangle, an angle bisector is a line segment or ray drawn from a vertex that cuts the angle in half.
In a triangle, a perpendicular bisector of a side is a line drawn perpendicular to a side of the triangle through its midpoint.
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
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.
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.
To support his claim that the line of reflection is a median of the equilateral triangle, Travis will need to show that:
To support her claim that the line of reflection is an angle bisector of the equilateral triangle, Tehani will need to show that:
To support her claim that the line of reflection is an altitude of the equilateral triangle, Clarita will need to show that:
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.
Use four different colors to identify the statements each of the students—Travis, Tehani, Clarita, and Carlos might use to make their case.
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
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
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.
Equilateral triangles can be folded onto themselves about a line of reflection
Definition of reflection
Definition of midpoint
Definition of median
Write each of Clarita’s, Tehani’s, and Carlos’s arguments in two-column proof format.
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.
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
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
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.
Identify each quadrilateral as a parallelogram, rectangle, rhombus, square, or none of these.
none of these
none of these
none of these
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.