Lesson 6 Claims and Conjectures Solidify Understanding

Jump Start

You created the following diagram previously, by rotating a triangle about the midpoint of one of its sides.

Today we are going to extend this diagram until it fills the paper. This type of plane-filling diagram is called a tessellation. Tessellations are diagrams created by a sequence of repeated figures produced by translations, rotations, or reflections that completely fill a plane with no gaps or overlaps.

Start by cutting out a scalene triangle from the 3x5 card. The longest side should be less than 3 inches. Use the ruler to draw the sides so they will be straight line segments.
Use the ruler to locate and mark the midpoint of each side of your triangle. Color each of the three angles a different color.
Trace your triangle in the middle of the white piece of paper. Color corresponding angles on the triangle you traced to match the angles on the triangle you will rotate.
Build your tessellation by rotating the triangle around the midpoint of one of its sides, tracing the triangle in its new position and color-coding the angles. Continue to rotate, trace, and color multiple triangles until your diagram begins to fill the page horizontally and vertically.

Learning Focus

Make conjectures about vertical angles and exterior angles of a triangle by reasoning with a diagram.

Make conjectures about angles formed when a line intersects two or more parallel lines by reasoning with a diagram.

Tessellations are diagrams created by a sequence of repeated figures produced by translations, rotations, or reflections that completely fill a plane with no gaps or overlaps. What do tessellations reveal about relationships between the angles formed when a line intersects two or more parallel lines?

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, we will make some conjectures about lines, angles, and triangles and in the next task write proofs to convince ourselves that our conjectures are always true.

First, we need to name and define some new angles that occur in this diagram. Read the descriptions of each of these angles as given in the task and then record the definitions of these angles in your own words.

Vertical Angles

When two straight lines intersect, the adjacent angles form linear pairs of angles because they are supplementary angles (that is, two angles whose measures sum to $180 °$ ). The opposite angles formed at the point of intersection are called vertical angles. In the diagram, $∠ 1$ and $∠ 3$ form a pair of vertical angles, and $∠ 2$ and $∠ 4$ form another pair of vertical angles.

1.

Examine the tessellation diagram, looking for places where vertical angles occur. (You may have to ignore some line segments and angles in order to focus on pairs of vertical angles. This is a skill we have to develop when trying to see specific images in geometric diagrams.)

Based on several examples of vertical angles in the diagram, write a conjecture about vertical angles.

My conjecture:

Exterior Angles of a Triangle

When a side of a triangle is extended in a straight line, as in this diagram, the supplementary angle formed on the exterior of the triangle is called an exterior angle. Note that the exterior angle and the adjacent interior angle form a linear pair of angles. 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.

Examine the tessellation diagram, looking for places where exterior angles of a triangle occur. (Again, you may have to ignore some line segments and angles in order to focus on triangles and their exterior angles.) Based on several examples of exterior angles of triangles in the diagram, write a conjecture about exterior angles.

My conjecture:

Parallel Lines Cut by a Transversal

When a straight line intersects two or more other straight 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 below, $∠ 1$ 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.

Examine the tessellation diagram, looking for places where parallel lines are crossed by a transversal line. Based on several examples of parallel lines and transversals in the diagram, write some conjectures about corresponding angles, alternate interior angles and same side interior angles.

My conjecture:

Ready for More?

Justifying Our Conjectures

In the next lesson, you will be asked to write a proof that will convince you and others that each of the conjectures you wrote above is always true. You will be able to use ideas about transformations, linear pairs, congruent triangle criteria, etc. to support your arguments. A good way to start is to write down everything you know about the diagram and then identify which statements you might use to make your case. To get ready for the next lesson, revisit each of the conjectures you wrote about and record some ideas that seem helpful in proving that the conjecture is true.

Takeaways

Based on reasoning with a diagram, we have surfaced the following conjectures:

Vertical Angles

Conjecture:

Exterior Angles of a Triangle

Conjecture:

Parallel Lines Cut by a Transversal

Conjecture:

Adding Notation, Vocabulary, and Conventions

Vertical angles:

Exterior angle of a triangle:

Remote interior angles of a triangle:

Transversal:

Adjacent angles:

Straight angle:

Linear pair of angles:

Supplementary angles:

Angles formed by parallel lines intersected by a transversal.

Alternate interior angles:

Corresponding angles:

Same-side interior angles:

Vocabulary

Lesson Summary

In this lesson, we used a colored-coded tessellation diagram to identify conjectures about relationships between a variety of different sets of angles, including: vertical angles, which are formed by the intersection of two lines; exterior angles of a triangle, which are formed by extending a side of a triangle; and the angles formed when two parallel lines are intersected by a line called a transversal.

Retrieval

Based on the diagram, provide justification for each of the statements.

1.

$\overset{―}{A B} ≅ \overset{―}{B D} ≅ \overset{―}{D E} ≅ \overset{―}{E A}$

2.

$△ B A E ≅ △ B D E$

3.

$∠ E B D ≅ ∠ E B A$

In a previous course, you studied absolute value functions. Use the given equation to sketch a graph of each function.

4.

$f (x) = | x - 3 | + 1$

5.

$g (x) = | x + 5 | - 3$