# Unit 7Congruence, Construction, and Proof

## Lesson 1

### Learning Focus

Construct a rhombus, a perpendicular bisector, and a square using only a compass and a straightedge (unmarked ruler) as tools.

### Lesson Summary

In this lesson, we learned about constructions: creating geometric figures precisely, using only a compass and a straightedge. Using only these tools, we constructed a rhombus with a given side and angle, constructed the perpendicular bisector of a side, and constructed a square with a given right angle and line segment for a side. We learned the value of the definition of a circle—the set of all points in a plane equidistant from a fixed center point—since circles allow us to construct congruent line segments.

## Lesson 2

### Learning Focus

Construct parallel lines and inscribed regular polygons.

### Lesson Summary

In this lesson, we learned how to construct regular hexagons and equilateral triangles by inscribing them in a circle. We also learned how to construct a line parallel to a given line through a given point. These constructions were based on properties of figures that we have observed in previous lessons.

## Lesson 3

### Learning Focus

Show two figures are congruent based on an efficient and consistent sequence of rigid transformations.

### Lesson Summary

In this lesson, we explored a sequence of rigid transformations that could be used to demonstrate that one geometric figure is congruent to another. While many sequences can be found, one particular sequence was identified as being more consistent and easy to replicate each time we need to show that two figures are congruent.

## Lesson 4

### Learning Focus

Explore and justify triangle congruence criteria using rigid transformations.

### Lesson Summary

In this lesson, we learned that it is not necessary to know that all pairs of corresponding angles and sides are congruent before we can claim that two triangles are congruent. There are several conditions where three pieces of information about corresponding congruent parts of the two triangles are sufficient to guarantee congruence of the triangles. We were able to justify the triangle congruence criteria by relying on the properties of rigid transformations to preserve distance and angle measures.

## Lesson 5

### Learning Focus

Identify congruent triangles, and write congruency statements.

Use triangle congruence criteria to justify other properties of geometric figures.

### Lesson Summary

In this lesson, we found several strategies for justifying statements about congruence by first looking for congruent triangles within a geometric figure. One helpful strategy was adding auxiliary lines to the figure. Another was to use the idea that corresponding parts of congruent triangles are congruent.

## Lesson 6

### Learning Focus

Justify construction strategies.

### Lesson Summary

In this lesson, we examined some standard constructions, such as bisecting an angle, constructing a line perpendicular to a given line through a given point, and constructing a line parallel to a given line through a given point. We found that we could explain why these constructions work based on properties of quadrilaterals, using corresponding parts of congruent triangles, or relying on the defining features and properties of rigid transformations.