Lesson 7 Justification and Proof Practice Understanding

Jump Start

Given conditional statement: If it is raining, the sidewalks are wet.

1.

Write the converse of the conditional statement:

2.

Assuming that the given conditional statement is true (that is, the sidewalks are exposed to the rain), is the converse a true statement? Explain why or why not.

Learning Focus

Practice translating proof-ideas into written formats.

What do I need to attend to when I write a formal proof?

What format should I use: narrative paragraphs, flow diagrams, two-column format? How does each format support my thinking?

What understandings might I draw upon: rigid transformations, triangle congruence, algebra?

Open Up the Math: Launch, Explore, Discuss

The diagram from “How Do You Know That?” has been extended by repeatedly rotating the image triangles around the midpoints of their sides to form a tessellation of the plane, as shown.

Using this diagram, you have made some conjectures about lines, angles, and triangles. In this task, you will write proofs to convince yourself and others that these conjectures are always true.

For each of the following proofs, you may use any format you choose to write your proof: a flow proof diagram, a two-column proof, a narrative paragraph, or an algebraic proof.

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.

1.

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

Prove: $∠ A E B ≅ ∠ C E D$

Exterior Angles of a Triangle

When a side of a triangle is extended, as in the diagram below, the angle formed on the exterior of the triangle is called an exterior angle. The two angles of the triangle that are not adjacent to the exterior angle are referred to as the remote interior angles. In the diagram, $4$ is an exterior angle, and $1$ and $2$ are the two remote interior angles for this exterior angle.

2.

Given: $∠ 4$ is an exterior angle of the triangle.

Prove: $m ∠ 4 = m ∠ 1 + m ∠ 2$

Parallel Lines Cut by a Transversal

When a line intersects two or more other lines, the line is called a transversal line. When the other lines are parallel to each other, some special angle relationships are formed. To identify these relationships, we give names to particular pairs of angles formed when lines are crossed (or cut) by a transversal. 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.

3.

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

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

4.

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

Prove: Alternate interior angles $∠ 3$ and $∠ 6$ are congruent.

5.

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

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

6.

What strategies seem useful for getting started on a proof?

Ready for More?

Prove the converse of the alternate interior angles theorem and the corresponding angle theorem:

1.

Given: Alternate interior angles $∠ 3$ and $∠ 6$ are congruent.

Prove: $\overset{\leftrightarrow}{E F} ∥ \overset{\leftrightarrow}{C D}$

2.

Given: Corresponding angles $∠ 2$ and $∠ 6$ are congruent.

Prove: $\overset{\leftrightarrow}{E F} ∥ \overset{\leftrightarrow}{C D}$

Takeaways

Today we proved the following theorems:

When a transversal cuts across parallel lines,

In addition, we proved the following converse statements as theorems:

My strategies for getting started on a proof include:

The strategies of my peers include:

Vocabulary

converse statement
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we practiced writing proofs for conjectures that we have explored in the past. We proved several useful theorems about the relationships between angles formed by two parallel lines and a transversal. Like triangle congruence criteria, these theorems about parallel lines will be useful in future proofs.

Retrieval

Use a compass and straightedge to construct each figure.

1.

Construct a rhombus using $\overset{―}{A B}$ as one of the sides and the given angle as one of the angles.

2.

Construct the bisector for the line segment and construct the bisector of the angle.

a.

b.

3.

Find the value of $x$ and the measure of the missing angles.