Lesson 3 Viable, Valid, Verified Solidify Understanding

Learning Focus

Examine characteristics of valid proofs.

What make an argument viable (workable, feasible, practical) and valid (a sound argument based on logic or fact)?

Are there different ways that logic reasoning and valid proof can be provided for the same theorem?

Open Up the Math: Launch, Explore, Discuss

In the geometry units of NC Math 2 you used geometric reasoning to help you to make sense of geometric situations and principles and wrote geometric proofs to help others follow your reasoning and be convinced you had made a sensible argument. One important skill for all students of mathematics is that of creating viable arguments and critiquing the reasoning of others.

Additionally, in NC Math 2 you discussed the “ways of knowing” framework, which described four common ways people come to accept something as true. You may recall the four ways of knowing are:

Accepting it on authority
Trying it out with multiple examples
Making an argument based on a diagram or representation
Making an argument based on previously proven statements and logical ways of reasoning

Your work in this task is related to the last of the four ways of knowing: proving statements. You will determine the validity of an argument based on the logical reasoning involved! Although you might use some of the other ways of knowing to think and reason about a situation, it is not considered mathematical proof until logical reasoning is applied.

For each of the following situations, there are several potential attempts to provide a logical proof by students. Your job is to determine if the reasoning is viable and valid. You need to decide if the argument makes sense, if the statements flow and connect, and if the argument is based on logic or some other type of reasoning. In essence, it is your job to verify the reasoning!

Several theorems you saw in the NC Math 2 course are presented below, along with a few samples of students’ attempts to prove these statements. For each proof provided, work with a partner using the following prompts to help you determine if the argument is precise and complete, or how to revise and refine the justification, if necessary:

Look carefully at the words and diagrams. Do they make sense?
Do we see a mistake that needs to be fixed?
Has something been left out that needs to be put in?
Do we know something more that needs to be added?
Why do we want to make the changes we are proposing?

1.

Vertical Angles

When two lines intersect, the opposite angles formed at the point of intersection are called vertical angles. In the diagram, $∠ A E B$ and $∠ C E D$ form a pair of vertical angles.

Given: $\overset{\leftrightarrow}{A C}$ and $\overset{\leftrightarrow}{B D}$ intersect at $E$ .

Prove: $∠ A E B ≅ ∠ C E D$

a.

Student A:

Statements	Justification
$\overset{\leftrightarrow}{A C}$ and $\overset{\leftrightarrow}{B D}$ intersect at $E$	Given
$m ∠ B E C = m ∠ A E D$	Vertical angles are congruent
$m ∠ A E B + m ∠ B E C = 180 °$	Definition of supplementary
$m ∠ C E D + m ∠ A E D = 180 °$	Definition of supplementary
$∠ A E B ≅ ∠ C E D$	Must be congruent based on algebra

Valid

Not Valid

Reason:

b.

Student B:

So, I see that it is given that $\overset{\leftrightarrow}{A C}$ and $\overset{\leftrightarrow}{B D}$ intersect at $E$ . This means that we have straight lines and when you rotate a line $180 °$ about a point on the line, it will land back on itself. So now I know I can rotate the given lines about point $E$ $180 °$ and they will land right back on top of themselves. By doing this $180 °$ rotation, $∠ A E B$ will land on $∠ C E D$ , which by definition means these two angles are congruent.

Valid

Not Valid

Reason:

c.

Student C:

Well, I can look at the visual provided and see that the angles are the same, and since this is asking us to prove they are the same, I think we are pretty much done. When you see the things you are trying to prove are true in a diagram, then you pretty much know that they are true without the diagram.

Valid

Not Valid

Reason:

d.

Student D:

Statements	Justifications
$\overset{\leftrightarrow}{A C}$ and $\overset{\leftrightarrow}{B D}$ intersect at $E$	Given
$m ∠ A E B + m ∠ B E C = 180 °$	Definition linear pair
$m ∠ C E D + m ∠ B E C = 180 °$	Definition linear pair
$m ∠ A E B + m ∠ B E C = m ∠ C E D + m ∠ B E C$	Substitution
$m ∠ A E B = m ∠ C E D$	Subtraction property of equality
$∠ B E C ≅ ∠ A E D$	Definition of congruence

Valid

Not Valid

Reason:

Pause and Reflect

2.

Parallel Lines Cut by a Transversal

When a line intersects two or more parallel lines, some special angle relationships are formed.

In the diagram, $∠ 1$ and $∠ 5$ are called corresponding angles, $∠ 3$ and $∠ 6$ are called alternate interior angles, and $∠ 3$ and $∠ 5$ are called same-side interior angles.

Given: $\overset{\leftrightarrow}{B F} ∥ \overset{\leftrightarrow}{A D}$

Prove: Corresponding angles $∠ 1$ and $∠ 5$ are congruent.

a.

Student A:

When you translate a line or any figure, we know by the properties and definition of the translation that all of the points move the same distance and direction. So, we can translate $\overset{\leftrightarrow}{B F}$ such that point $B$ lands on point $A$ and when we do this, the lines $\overset{\leftrightarrow}{B F}$ and $\overset{\leftrightarrow}{A D}$ will land on top of each other because by definition of parallel, they are the same distance apart. By doing this, we will end up with angles that correspond like $∠ 1$ and $∠ 5$ as well as all of the other corresponding angles landing on top of one another: $∠ 2$ and $∠ 6$ , $∠ 3$ , and $∠ 7$ , etc. So, corresponding angles are congruent.

Valid

Not Valid

Reason:

b.

Student B:

Well, we just did a proof about vertical angles and so I know that they are congruent. From the given, I know we have lines that intersect, so I can use the vertical angles to show $∠ 1 ≅ ∠ 4$ . I also know that rotations of $180 °$ cause lines to land on themselves and keep distances from the center of rotation equal. So, if $\overset{―}{A B}$ rotates around its midpoint $180 °$ , it will land back on top of itself and the parallel lines will land on top of each other. This will put $∠ 4$ on top of $∠ 5$ , so $∠ 4 ≅ ∠ 5$ . We now have $∠ 1 ≅ ∠ 4$ and $∠ 4 ≅ ∠ 5$ , so by substitution, $∠ 1 ≅ ∠ 5$ .

Valid

Not Valid

Reason:

c.

Student C:

Statements	Justifications
$∠ 1 ≅ ∠ 4$	Vertical Angles
$∠ 2 ≅ ∠ 3$	Vertical Angles
$∠ 6 ≅ ∠ 7$	Vertical Angles
$∠ 5 ≅ ∠ 8$	Vertical Angles
$∠ 1$ and $∠ 5$ are congruent.	Conclusion

Valid

Not Valid

Reason:

d.

For this, I took a protractor, you know the thing that you can measure angles with, and I measured a bunch of the angles and they matched. I drew and labeled more diagrams a little different from this one, and when I measured them, they matched, and so I know that they will be congruent when this happens.

Valid

Not Valid

Reason:

Ready for More?

Given: $\overset{\leftrightarrow}{B F} ∥ \overset{\leftrightarrow}{A D}$

Prove: Same side interior angles $∠ 3$ and $∠ 5$ are supplementary.

You may use the theorem proved previously in this task that corresponding angles are congruent. For example: $∠ 1 ≅ ∠ 5$

Takeaways

A complete and logical mathematical proof consists of a logical sequence of statements justified by

The geometric reasoning in a proof can be based on

Examples of geometric reasoning that support the proof-process, but are not sufficient as proof, include

Vocabulary

reasoning – deductive/inductive
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we reviewed what it means to write a valid proof, and we examined examples of proofs that were written in different formats but demonstrated the characteristics of logical reasoning that is the hallmark of valid proof. We also examined other examples that were problematic in different ways. Discussing these examples will help us to be more thoughtful and strategic when writing proofs. The proofs we reviewed today were about vertical angles and the angles formed when parallel lines are crossed by a transversal.

Retrieval

1.

Construct a parallelogram using $\overset{―}{B C}$ as one of the sides and point $A$ as one of the vertices.

2.

Graph the quadratic functions and state the vertex of the graph.

a.

$g (x) = {(x + 5)}^{2} - 3$

Vertex?

b.

$f (x) = - 2 {(x - 3)}^{2} + 4$

Vertex?