# Unit 1Congruence, Structure, and Proof

## Lesson 1

### Learning Focus

Identify the defining features of the translation, rotation, reflection, and dilation transformations.

Use function notation to describe transformations.

### Lesson Summary

In this lesson, we reviewed the defining characteristics of each of the three rigid transformations, which preserve angle and distance measurements from pre-image to image. We also reviewed the characteristics of the dilation transformation, which produces similar figures. We also examined notation for describing these transformations symbolically. The symbolic notation illustrates that geometric transformations are functions, with each set of points in the input image being mapped to a unique set of points in the output image.

## Lesson 2

### Learning Focus

Justify the triangle congruence criteria using reasoning based on rigid transformations.

### Lesson Summary

In this lesson, we reviewed triangle congruence criteria that guarantee two triangles are congruent without having to know that all corresponding sides and corresponding angles are congruent. We justified each set of triangle congruence criteria using rigid transformations and reviewed what it means to justify a claim.

## Lesson 3

### Learning Focus

Examine characteristics of valid proofs.

### 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.

## Lesson 4

### 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 5

### 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 6

### 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.