# Unit 3Geometric Figures

## Lesson 1

### Learning Focus

Examine ways of knowing that the sum of the angles in a triangle is .

### Lesson Summary

In this lesson, we explored different ways of knowing if something is true, such as basing our knowledge on being told by someone in authority, versus basing our knowledge on experimentation or reasoning with a diagram. We examined these ways of knowing in the context of justifying how we know that the sum of the angles in a triangle is .

## Lesson 2

### Learning Focus

Examine features of diagrams to determine the story of how the diagrams were built.

Write a paragraph to prove a conjecture that surfaces when analyzing a diagram.

### Lesson Summary

In this lesson, we learned that diagrams are built in consecutive steps, each step depending upon previous steps. When we can tell the story of how the diagram was built, we can also identify additional features that might be true about the diagram and gain insight into how to prove these new conjectures. Today we wrote paragraph proofs about the things we noticed in a diagram to verify that our observations were true.

## Lesson 3

### Learning Focus

Organize and sequence proof statements using a two-column format.

Examine a claim about points on a perpendicular bisector of a line segment.

### Lesson Summary

In this lesson, we learned how to write proofs in a two-column format. This format helps us keep track of the logical organization and sequence of statements in a proof so each statement in the proof can be justified based on the statements that come before it. We proved a statement about the points on a perpendicular bisector of a segment and a statement about the base angles of an isosceles triangle.

## Lesson 4

### Learning Focus

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.

### Lesson Summary

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.

## Lesson 5

### Learning Focus

Determine when a line will be parallel to its pre-image after a translation, rotation, or reflection.

Build a system of geometry based on definitions, postulates, and theorems.

### Lesson Summary

In this lesson, we learned that a system of geometry includes definitions, postulates, and theorems. We developed postulates for parallelism for each of the rigid transformations based on experiments that helped us determine under what conditions corresponding pre-image and image lines would always be parallel following a translation, a rotation, or a reflection.

## Lesson 6

### 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.

### 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.

## Lesson 7

### Learning Focus

Practice translating proof-ideas into written formats.

### 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.

## Lesson 8

### Learning Focus

Use theorems about the relationships of angles formed by parallel lines and a transversal to prove properties of parallelograms.

### Lesson Summary

In this lesson, we drew upon our understanding of rigid transformations, triangle congruence criteria, and postulates and theorems about parallel lines to prove many of our conjectures about the sides, angles, and diagonals of parallelograms.

## Lesson 9

### Learning Focus

Classify and justify types of parallelograms based on characteristics of their angles and diagonals.

### Lesson Summary

In this lesson, we expanded our ways of thinking about proofs by starting with statements where we first had to decide what we were given and what we were trying to prove. We had to create our own diagrams and mark congruent parts as they became apparent to us based on reasoning with a diagram. We noticed that carefully sequencing proofs allows us to draw upon some theorems to prove other, more complicated theorems.

## Lesson 10

### Learning Focus

Examine properties of the medians, angle bisectors, and perpendicular bisectors of the sides of triangles.

Construct the center of a circle that will pass through all three vertices of a triangle and the center of a circle that can be drawn in a circle so that it touches all three sides.

Construct the “balancing point” of a triangle.

### Lesson Summary

In this lesson, we found that the three medians of a triangle are concurrent, which means that they all meet at the same point on the interior of a triangle. We also found that the three angle bisectors of a triangle are concurrent, as well as the three perpendicular bisectors of the sides. These points of concurrency are called centers of a triangle, since they locate interesting points in the interior or exterior of the triangle, such as the balancing point or the centers of circles that can be drawn to pass through all three vertices of the triangle or touch all three sides.