Lesson 7 Justification and Proof Practice Understanding

Jump Start

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


Write the converse of the conditional statement:


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.

A repeated pattern of a triangle with a green arc, blue arc, and red arc for each angle.

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, and form a pair of vertical angles.

Line BD and Line AC intersect at point E.


Given: intersect at .


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, is an exterior angle, and and are the two remote interior angles for this exterior angle.

A triangle with angles labeled 1-3. One side of the triangle extends forming exterior angle 4 that makes a linear pair with angle 3.


Given: is an exterior angle of the triangle.


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, and are called corresponding angles, and are called alternate interior angles, and and are called same side interior angles.

Transversal line AB intersect lines EF and CD forming angles 1-8. 12345678



Prove: Corresponding angles and are congruent.



Prove: Alternate interior angles and are congruent.



Prove: Same-side interior angles and are supplementary.


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:


Given: Alternate interior angles and are congruent.


Transversal line AB intersect lines EF and CD forming angles 1-8. Point M lies on line AB. 12345678


Given: Corresponding angles and are congruent.



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:


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.


Use a compass and straightedge to construct each figure.


Construct a rhombus using as one of the sides and the given angle as one of the angles.

Angle A with one side line segment AB and the other ray A.


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


A line segment.


An angle.


Find the value of and the missing angles.

Also find the value of angle .

A triangle with angles labeled 2x-18, 3x-12, and 40 degrees. Exterior angle B forms a linear pair with the angle labeled 40 degrees.